An Old School Dogfight is a type of close-combat battle between fighter craft where each attempts to maneuver behind the other in order to shoot them down. In essence, it's a three dimensional version of Chasing Your Tail, where both parties are armed with a Fixed Forward Facing Weapon, making it necessary to be behind their target in order to attack. In Real Life, this form of combat is limited to fighter planes, but fiction is less restricted — it can show up not only with Cool Planes, but Space Fighters, submarines, and even tunneling vehicles like Drill Tanks. Any time a 3-D environment is combined with Fixed Forward Facing Weapons, you can expect to run into old school dogfighting.
The reason it's called an "old school" dogfight is because it doesn't actually happen very often in real life anymore. While it was the standard form of air-to-air combat for decades, the introduction of reliable, long range guided missiles means that most modern air combat is now fought well beyond visual range. Old school dogfights can still involve the use of missiles, though — they'll just be much shorter ranged than their real life equivalent, and have to be fired from behind in order to have a chance of hitting.note This wasTruth in Television for certain types of missiles, but isn't anymore.
The characters involved in an Old-School Dogfight will frequently be Ace Pilots with Improbable Piloting Skills, often have a Wingman and/or a Guy In Back, and will probably cover each other's six at some point. Old school dogfighting during an Airstrike Impossible is not out of the question. Overlaps with Space Is Air when done by Space Fighters.
Examples
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Anime and Manga
Cowboy Bebop features old school dogfighting in several episodes, as well as the movie. The series normally featured realistic space flight, but used this trope during space combat because it's cooler.
Fighters in Space Battleship Yamato use tactics identical to World War II-era carrier aircraft, including dive bombers and torpedo bombers. Very much a part of the stylistic "feel" of the show, with the very first episode (heavily edited when it first aired in the US) even including the original WWII battle where the Yamato got sunk (by air attack) to illustrate the point.
Valkyries normally use missiles — lots and lots of missiles — leading the series to be known for the "Itano Circus", a stylized depiction of close-range dogfighting that includes the use of missiles.
Macross Frontier actually plays it straight, subverts it, and inverts it at some points. Most of the time it's played straight as it is in the other series, with two notable exceptions. First, the Ghost V-9s behave very much like they're in space, changing direction with zigs and zags, which is where it's subverted (since everything else up to that point played it straight). Second, earlier in the series, Brera dodges a gun burst from Alto by zagging sideways in an atmosphere, giving the appearance of actually inverting the trope.
Film
The Lost In Space movie opened with a classic space dogfight.
Star Wars was the Trope Codifier for space fightersbehaving this way, to the point where Luke's experience flying a skyhopper (a mostly atmospheric ship, though capable of suborbital travel) helps him fly an X-Wing. Lucas even used old movies to choreograph the space battles. The Death Star assault scene was modeled after the film The Dam Busters (1955). In addition, the sequence was partially inspired by the climax of the film 633 Squadron (1964) in which RAF Mosquitos attack a German heavy water plant. Clips from both films were included in Lucas's temporary dogfight footage version of the sequence.
The novelizations of the Robotech series address this problem by noting that the Veritech fighters were partially controlled by the pilot's thoughts. And since the pilots were accustomed to flying in the atmosphere, that translated to similar flying patterns in space. Whether this makes it better or worse is up for debate.
The Gaunt's Ghosts spinoff Double Eagle, hyper advanced space fighters with vectored thrust engines and laser cannons are still mixing it up WWII style.
On the other hand, mocked in Ciaphas Cain: Death or Glory along with See the Whites of Their Eyes: "Contrary to what you might see in an episode of Attack Runfootnote A popular holodrama of the 930s, about a squadron of fighter pilots in the Gothic War., starships in combat seldom approach to within point blank range of one another, exchanging fire at distances of hundreds, if not thousands, of kilometres." We should note here that an Ork space fighter later attacks Cain's Escape Pod in the stereotypical style, but like most Orky silliness this is probably justified by their inborn Clap Your Hands If You Believe field allowing the fighter to do that.
Played with for what little we see of fighter combat in The Flight Engineer (the series revolves around a space carrier, but the hero is the fighter squadron's chief engineer instead of a pilot). Speeds chase each other's tails and loop around crazily and are armed with line-of-sight weapons, but the ranges are far greater than usual and, given the fact that capital ships avert Space Friction (they have to flip end-for-end and fire thrusters to stop), it's possible the Speeds were specifically designed that way.
Live-Action TV
Although Space Is an Ocean in the Star Trek franchise, there are occasional space battles that resemble dogfights more than naval battles, particularly as the budgets and special effects technology has increased. Later episodes and films show starships and starfighters alike banking, rolling, and chasing one another's tails — and not only that, but they fight at ranges comparable to WWII-era dogfights. Certain ships appear to be specifically designed for dogfighting; Klingon Birds-of-Prey and Jem'Hadar fighters, for example, only have Fixed Forward Facing Weapons, and while Federation capital ships can fire their phaser strips in nearly any direction the Defiant-class also has fixed phaser cannons that are more powerful than its strips.
Farscape opens with John getting shot into the middle of an old school dogfight.
The Stargate Verse has a few dogfights, and every race except the Asgard has at least one vehicle useful in a dogfight (F-302s, Death Gliders, Puddle Jumpers, Wraith Darts etc.). The F-302 and Death Gliders are hybrid aircraft that can also fight in atmosphere, which might excuse their aerodynamic design to some degree. Unfortunately this falls flat in light of Puddle-Jumpers and Darts, neither of which are all that aerodynamic, but both of which can not only fight in atmosphere, but travel through Stargates. The Goa'uld also invented a Stargate traveling Death Glider at one point, but due to the difficulty of "threading the needle" only master pilots were able to use them for that purpose. Also the F-302 originally used modified Sidewinder missiles instead of line-of-sight weapons, but by later Stargate Atlantis episodes they were replaced with railguns.
Space: Above and Beyond uses this trope, though the Human SA-43 "Hammerhead" fighters had a pair of turreted guns (one in front and one behind) so they could shoot at targets in any direction.
The interceptors in UFO use stand-off missiles, though they appear to be unguided. The combat in general draws very much from popular Battle of Britain images: Moonbase is the beleaguered sector airfield and SID (Space Intruder Detector) the RDF radar. Calmly-speaking young women (the WAAF's) vector in Interceptors (Spitfires) against the anonymous alien invaders (German bombers). But given that the Moon takes 27.322 days to orbit the Earth, one wonders why the aliens don't just attack when Moonbase is on the opposite side of their target.
Doctor Who came up with an example of this trope taken to its extreme: through some Technobabble upgrades, 1940's Spitfires take on a Dalek ship in orbit.
Babylon 5 and its spin-off Crusade play this straight. Even ships much larger than fighters can engage in such behavior, such as the White Stars, which are classified as gunships.
Tabletop Games
Played straight in Battle Tech standard rules. Aerospace fighters function much the same in either space or atmosphere — there are some practical differences between the environments, but the basic maneuvers and tactics are still the same. The advanced rules (found in the Strategic Operations volume in the current incarnation of the rule set), though, suddenly has fighters in space that can use their maneuvering thrusters to spin in mid-flight in ways that they never could in mid-air. (This doesn't change their heading, mind, just their facing — but it still effectively turns them into fast-moving turrets.)
The Imperium has access to Skystrike missiles for air-to-air combat, and Tau seeker missiles give them a distinct edge over other races, but for the most part the game plays this trope straight.
This trope is the entire purpose of the spinoff game Aeronautica Imperialis. The only guided missiles in the game are ground-attack-only weapons.
Video Games
A number of combat flight simulator games featuring missile-armed jets fiddle with this trope.
In Janes Advanced Strike Fighters, for instance, some dogfighting skill is required, but you usually engage the enemy from several kilometers away. Most kills will come as you close with other planes, but regardless of how you kill you can only get a missile lock when the enemy is ahead of you.
The Ace Combat series Zig-Zags this trope; on the one hand, the player character's plane is typically armed with dozens of missiles that they can use to blow away mooks by the hundred, but only at absurdly short ranges, and ace pilots or Boss Battles generally will require some old school dogfighting to take down.
Tom Clancy's HAWX uses the Ace Combat model for the most part and even forces you to guns only at one point (handwaved by a malfunction in your squadron's missile targeting systems). It also features a mechanic where you can turn off your plane's stall-prevention systems, enabling you to do such things as flipping end-for-end to blast a guy behind you.
Star Wars games, as in the rest of the franchise, make extensive use of old school dogfighting.
In Ace Online, while ridiculously high-altitude and long-range missile combat is possible, war situations tend to move towards close-range combat for numerous reasons, including:
A-gears in Siege Mode are capable of killing at extreme distances due to the volume of firepower they're capable of putting out. However, their weapons have a minimum range on their weapons (which is affected by their armor), meaning that the best way to kill them is to — you guessed it — old school dogfight.
B-gears are able to rain death from the very top of the battlefield. However, they're often used instead as a battering ram in nation war events, in which they attempt to destroy defenders pouring weapons fire into a zone gate with an Action Bomb that takes out nearby gears.
I-gears have amazing maximum lock-on range with their missiles, but it's also harder to lock on to the desired target from extreme range if said target is surrounded by clouds of sacrificial decoy craft.
Used in the indie space fighter game Critical Mass, in which your squad does indeed have guided missiles, but the ships often turn faster than the missiles do. Add this to the fact that a lot of ships are equipped with the powerful but unguided Plectron rocket, and you get old school dogfighting.
X-Universe: Used like pretty much everything else set in space. X3: Reunion is a particularly egregious offender, as it advertised a Newtonian physics model but then still went with the constant thrust = constant speed model. Most medium to heavy missiles are not only easily shot down by guns, but are actually slower than their intended targets. Arguably a response to Artificial Stupidity, as the missiles are prone to Roboteching and there's no such thing as countermeasures.
Star Fox is built on this trope, which shouldn't be surprising since the ships used are basically just the X-wings from Star Wars. Levels take place in open space, near the ground, or even near the surface of a sun, but the handling is always exactly the same.
Halo Reach has a Wing Commander-style space dogfight in a Sabre fighter during the mission "Long Night of Solace", armed with machine guns and barrages of homing missiles. Several levels in the Halo franchise also have Banshee dogfighting, where they handle the same regardless of whether they're on Earth, a Halo ring, a gas giant, or in space.
In Tachyon: The Fringe, all engagements are done at extremely close ranges, and standard tactics involve chasing the enemy until your have him in your crosshairs for a (slow) laser barrage. While there are missile weapons, they are limited, can be fooled with cheap countermeasures, and only those provided by GalSpan are guided.
Mass Effect: Sometimes averted, usually not. The codex averts it: point-blank range, where fighters can attack, is tens of kilometers and most battles supposedly take place at hundreds or even thousands. However Mass Effect 2 shows the Normandy SR-2 battling some Attack Drones this way using its point-defense lasers, and Mass Effect 3 has a few occasions of Citadel fighters doing it against drones and Reaper Harvesters. The big one being the battle over Earth, which has a huge dogfight between Alliance starfighters and Reaper drones, with dogfighting using the stereotypical ranges and maneuvers and the capital ships doing the Standard Starship Scuffle in the background. Clearly the cutscene artists didn't read the codex.
Webcomics
Crimson Dark has space fighters and bombers which act like planes.
Angels 2200 is about the pilots of carrier-based space fighters which look and fly like planes.
Real Life
Old school dogfighting was the only form of air-to-air combat up until the development of reliable BVR (beyond visual range) and all-aspect (capable of hitting a target at any angle, not just one that's in front of you) missiles. Despite advances in technology, however, old school dogfighting never entirely disappeared. There was a period in the early 1960s where dogfighting was briefly considered obsolete, and planes (notably the American F-4 Phantom) were designed with missiles as their only weapon. The Vietnam war demonstrated the problems with this approach, and pretty much every plane designed for air-to-air combat since then has planned for dogfighting to some extent.
As an analogy, dogfighting for pilots is like hand-to-hand combat for infantry. It's no longer the primary form of combat, and you don't really expect to use it on anything like a regular basis, but you still have to train for it, because if you don't, your enemies will use that against you.