No artificer or wizard can replicate the powerful artifacts found in Xen'drik. These objects wait to be found, and with international tensions already heightened by the Last War, possession of deadly magic weapons could easily shift the balance of power.
— Eberron, "Eberron Campaign Setting, Chapter 7: Life in the World"
In fiction there's often a lot of reasons why opposing factions want to develop better technology (although they only need one). While some just research and create it, others just find (or supplement their own creations with) Lost Technology. When both sides scramble to secure such caches you end up with this trope.
Where the technology's being salvaged from can vary greatly depending on the setting and story;
Cold War (and sometimes present day) settings often have abandoned Nazi technology (and scientists). Present day settings might also treat Soviet Superscience and scientists in the same fashion. More fantastic settings might provide a similar dynamic with a Fictional Counterpart: a recently defeated faction who've left lots of technology and research lying around.
Salvaging technology which was left behind by Precursors.
A variation might involve salvaging your own technology after a disaster of some sort destroyed your ability to manufacture more of it; either by digging it from the wreckage or finding caches. Or acquiring it from neutral factions who'll only hand it over to the first bidder.
Soul Eater Eibon's Demon Tools and the process of "keep away from Kishin" involving them. Specifically B.R.E.W and the Book of Eibon, and with good reason, as B.R.E.W is whatever you desire it to be, and the Book of Eibon is a collection of magical and dangerous creatures that can be summoned at any time by the user and actually IS Eibon himself at one point in the anime.
Orguss 02 is driven by this trope, with two rival nations excavating the Humongous Mecha from the original series and repairing them with Industrial-age weaponry. The plot thickens when the country our protagonist isn't from excavates an extremely Humongous Mecha powerful enough to win the war singlehandedly.
The Big O has both Giant Robots and other technology, collectively referred to as "Memories", due to the mass-amnesia accompanying the apocalyptic war in the series' backstory.
The Ancient Belka wars of the Lyrical Nanoha franchise was fueled by this and the Lensman Arms Race, with nations also scrambling to find various Lost Logia to give themselves an advantage over their enemies. The Cradle of the Saint Kings, for example, was not made by the Ancient Belkans, and was instead an especially powerful Lost Logia they discovered whose origins could be traced back to Al-Hazard.
Spriggan centers on the pursuit of OOPArts (Out-Of-Place Artifacts), ancient Lost Technology with the potential to disrupt the balance of power or even destroy the world if misused. Many governments and organizations are willing to resort to extreme measures to secure or destroy these artifacts.
This is alluded to in Raiders of the Lost Ark, since both the US government and the Nazis might want to use it as a weapon (although it's more of a powerful spiritual artifact than lost technology).
Colonel Musgrove: What's that supposed to be coming out of there?
Indiana Jones: Lightning. Fire. The power of God or something.
Major Eaton: I'm beginning to understand Hitler's interest in this.
Marcus Brody: The Bible speaks of the Ark laying waste to entire regions. An army which carries the Ark before it...is invincible.
A recurrent theme in some of Philip K. Dick's short stories, with post war survivors whose lives are dependent on ancient or alien technology and in some cases the will and resourcefulness needed to acquire them.
The galactic war described in Vladimir Vasilyev's The Legacy of Giants and No One but Us involves an attempt to recover a cache of Precursor portals that can be used to transport entire armadas instantaneously (normal FTL travel is fast but far from instantaneous and can be detected far in advance of arrival - arriving ships leave an "imprint" in space). The duology is even called "The War for Mobility", as both sides realize that the more mobile side wins. In the end, the good side (the one with humans) ends up obtaining the portals and quickly figuring out how to use them to capture the "evil" side's Emperor, forcing their surrender.
In The Stand, the good guys' main concern (once they've begun to rebuild, at least) is taking care of Randall Flagg before he starts utilizing all the old military weapons and equipment that are free for the taking now that The Plague has wiped out most of humanity. Flagg's minion Trashcan Man is particularly adept at finding these new toys and getting them in working order.
In Mortal Engines, as it is set in the future a long time after After the End, when civilization has rebuilt, most of the weapons technology comes from archeologists find "Old Tech" knowlege or superweapons.
Live Action Television
Babylon 5 - the humans are one of the least-advanced major galactic powers, dependent at first upon the Centauri for access to hyperspace jumpgates and hopelessly outclassed by the Minbari, who were on the verge of wiping humanity out. Even during the setting of the series proper humans are the only major power not to have developed artificial gravity. In response, the humans formed InterPlanetary Expeditions (IPX), a corporation that funds archaeological expeditions to locate and exploit the technology of the First Ones. The main plot kicks off when an IPX survey reaches Z'ha'dum and awakens the Shadows.
Revolution: More or less. One who can make pre-blackout tech work will have a gigantic advantage.
Stargate SG-1: There's a lot of this going on. Several Earth factions are desperate to get their hands on any alien technology in order to gain a political edge and also ensure survival against the Goa'uld threat. Meanwhile, all of the galactic factions (including Stargate Command) are scrambling to find any Ancient technology that might be left behind.
Played with by the fact that the dominant galactic power at the time the series begins, the Goa'uld, are scavengers by nature; virtually all of their technology, such as optical computers and even the Stargate, has been either found, stolen from someone else, or reverse engineered. This puts them at a disadvantage against more innovative species. But it also means that the handful of times the Goa'uld actually develop something new on their own, it tends to be extremely dangerous.
Star Trek: The Next Generation episode "The Chase" features several factions - Federation, Klingons, Cardassians and Romulans - who are all after a DNA code left by Precursors. Said code has been broken into several pieces, with various pieces in the possession of each of the aforementioned factions. The Klingons, in particular, think it's a weapon, while the Cardassians think it will yield an unsurpassed power source. It turns out to be a message to the Precursors' descendant species, talking about the commonality between them. Picard and the Romulan representative are receptive, the Klingons and Cardassians are not.
Star Trek: The Original Series episode "A Private Little War" was an allegory of the Vietnam War, with both factions being armed by humans and Klingons.
In the Warhammer 40,000 universe, this is often the case when different factions in the Imperium of Man (especially the Adeptus Mechanicus) end up fighting each other, as they've become technologically backwards and most of their best technology is either relics they've dug up (or stolen from each other) or created by the few xenos they haven't killed on sight. The Mechanicus also bans innovation as heresy against the ancients, forcing them to obtain all tech from archaeological relics built during the Dark Age of Technology.
Also occurs between species from time to time, particularly with ancient Necron/C'tan relics (often not known to be so by those trying to claim it). Particularly frequent with Orks, who will happily try to salvage and use anything regardless of who made it, what it's supposed to do or how dangerous it might be.
In BattleTech during the Succession War era, the discovery of Lostech caches can cause small-scale wars over their possesion. Then the Gray Death Legion mercenaries discovered the Memory Core with almost complete lostech schematics and spread it contents through the Inner Sphere despite the efforts of ComStar.
Implied in Eberron. Several countries are gearing up for war and there are a great deal of powerful magical artifacts to uncover. Whether the trope is played straight is up the DM, of course.
In the final days of the AlternityStar*Drive setting, the Galactic Concord and allies faced off against the Exeat. Each side tried to dig up artifacts left by various Precursors to use against each other. In the Battle of Aegis, the Exeat displayed such a weapon. It hit hard.
Exalted: All types of Exalts are in a race to unearth ancient Solar superweapons (and there are TONS of them), with the ones coming ahead being the Houses of the Dragon-Blooded, ready to use them in the inevitable civil war. The only ones who aren't in the race are Sidereals, whose interest is in keeping the artifacts buried.
In Sid Meierís Alpha Centauri mysterious monoliths, artifacts and other leftovers from the creators of Planet are scattered around the terrain and can boost scientific research of the colonist group who finds them first. Meanwhile, the alien factions have to slowly rediscover technology lost when their ships crashed on the surface after a firefight.
In the Halo universe, a theocratic alien Covenant is constantly searching for technology and structures left behind by the Forerunners (their religious fervour means that they consider them holy relics and won't modify them even slightly). The result is that Covenant energy weapons are barely better than conventional human firearms, but the Covenant still win because humans don't have force shield technology. When humanity starts getting curb-stomped, the UNSC (secretly) throws all of its researchers at the few Forerunner archaelogical sites it has. Halo 1 starts with the Pillar of Autumn deciding (on Cortana's whim) to warp-jump using Forerunner navigational coordinates so that the Covenant won't follow them to another human-inhabited planet.
The human side are more willing to reverse engineer what they can find, but most of the technology that gets put in the field is based on stolen Covenant tech. This is partially because the Covenant know where to look for Forerunner artifacts, but mostly because Forerunner tech is much more advanced and specialized than Covenant tech. As a result, Spartan armor is higher-quality than Elite armor (but Elites are still hard to kill because they are physically as strong as a Spartan in power armor).
In one of the books, Cortana hijacks a Covenant battle cruiser and discovers that its plasma cannon firmware is an absolute mess. She re-writes it in the middle of a space battle to convert the slow-charging mortar cannons into rapid-fire ion beams, then proceeds to tear apart several Covenant ships of the same caliber.
As soon as she starts messing with Covenant software, a Covenant AI jumps out and calls her a heretic for altering "holy relics". Yes, they deliberately stunt their AIs with the same religious restrictions.
As of Halo 4, humans have completely outpaced the Covenant in the technological arms race on all fronts. The situation is now reversed, with the Covenant looking for Forerunner technology that will let them defeat the UNSC.
The most advanced UNSC ship, the Infinity, is built by ONI and augmented using reverse-engineered Covenant and Forerunner tech (with the help of Engineers who personally knew Forerunners). Also, by this point, the SPARTAN program has reached a point where veteran soldiers can be converted into Super Soldiers without the horrendous attrition rate of the SPARTAN-II program, and the MJOLNIR armor can be produced much faster to accomodate them.
The entire conflict in Metal Fatigue is based around this; three brothers working for their Combot building family business discover alien technology, setting off a war between it and two other companies (one of which each of the brothers ends up working for). Naturally the best way to get the best parts is to find said alien technology (as well as stealing parts you've blased off enemy combots).
In Xenogears, most of the best Gears used in the war between Aveh and Kislev are dug up relics from previous wars, not newly manufactured. This naturally leads to conflicts over the best salvage sites.
Part of the setting of Panzer Dragoon. Several factions are after technology left behind by the Ancients, generally for this purpose:
The Empire is the largest and most notable of these factions. They were originally formed to try and help better society with the technology, but by the time the games take place, they have become corrupt.
In Saga, the Black Fleet (some of the Empire's most elite forces) defect, and claim a specific artifact (the woman Azel) to keep the Emperor's ambitions in check.
Another notable faction is the secretive group known as the Seekers. Commonly thought to be nothing more than outlandish tomb raiders, their goal is to use Ancient technology to help humanity reclaim the world from the various monsters and hazards. They are at odds with the Empire at the point the games take place, despite some initial common ground.
The Fallout games evoked this trope, as the Brotherhood of Steel is sometimes depicted as going to any length to secure pre-war weapon technology. Most such technology is found in Vaults, since everywhere else is a Scavenger World.
The first game in the series featured a nuke which you could use to skip the endgame boss fight.
In the second game of the series, the Enclave has access to enough fossil fuel to manage a small army, complete with helicopters. This made them more powerful than the Brotherhood of Steel, and they could curb stomp any faction in the Wasteland. They also genetically engineered a genocidal virus.
Fallout: New Vegas was particularly blatant about this, featuring a power plant which a fanatical former Brotherhood leader (hoping to dig up an ancient weapon system) defended to the bitter end, refusing to abandon the site despite his forces being massively outnumbered by the NCR.
Armageddon Empires, the post-apocalyptic strategy/card game. While all the factions are capable of researching new technology, most games also feature at least a few special tiles where you can dig up unique (and powerful) units, weapon systems, enhancements, and other goodies to unleash against your enemies.
The RTS flash game Relic Of War (previously "Reich of Darkness", but renamed and re-released due to some intolerants and paranoids censors) goes basically around this, with the nazis discovering an ancient artifact and using it for taking an advantage in an alternative timeline WWII developing new technologies (the allies do the same later on)
A good number of planets in Ascendancy feature ruin tiles that can be excavated, resulting in a random tech from the research tree. This can be anything from a tech you would've discovered the next turn from normal research to endgame tech that could give you a massive advantage. It normally takes 50 turns (25 if colony base is built on a red square) to build a dig site on a newly-colonized world.
A non-military version in the Brave New WorldExpansion Pack to Civilization V. After the discovery of Archaeology, you can start training Archaeologist units who can excavate antiquity sites for cultural artifacts that boost your Tourism rating. There are also hidden antiquity sites that are only visible to players who have maxed out the Exploration policy tree. These aren't any better than the regular sites, but, being the only one who can see and excavate them can give a huge advantage to somone going for a cultural victory. Also, other civs get annoyed when you start excavating within their borders, especially since excavating a site on an improved tile destroys the improvement.
The second season of Transformers Prime has the Autobots and Decepticons racing each other to retrieve the Iacon Relics - Cybertronian Lost Technology which was scattered across Earth for safekeeping.
In Real Life, during the Cold War both the West and the Soviet Union were quite keen to recruit former Nazi scientists (some of the same minds responsible for putting man on the moon also developed the V1 and V2 weapons which bombed London in WWII).
In the wake of the Cold War former Soviet technology and scientists were highly sought after by both the United States (in order to keep them out of the hands of other countries) and emerging world powers such as China and India (in order to bolster their own positions). The US and China were the #1 and #2 purchasers of ex-Soviet weapons systems in the world.
Most past societies considered relics to be extremely potent sources of power, either as a demonstration of a god's favor, or as literal spiritual weapons. As such, the discovery of relic caches could easily lead to local conflicts, as rival tribes feared the power such relics might bring to their enemies. Paradoxically though, this is actually more common the further back one goes through history, since tribal territories would occasionally become rich enough to produce or obtain works of a quality normally impossible for a given territory, only to later have that society collapse, encouraging neighboring tribes to fight for the now irreplaceable works.