Ryo and Kaori versus Union Teope. End result: Union Teope pulled out of Japan after Ryo and Kaori had killed most of their men, trolled into madness their best hitman before killing him, destroyed many of their assets and stolen a crapload of money.
The result of the second round? The Union's boss killed alongside many of their men, a crapload of drugs confiscated by the police, and Interpol got enough evidences and other informations to dismantle the entire organization.
In the first manga chapter/anime episode Ryo is hired to kill a boxer who, in order to become champion, had murdered the challenger (and fiancee of Ryo's client) and blackmailed the champion into throwing the fight. The awesomeness is not in Ryo killing the fool, but in how he did it: during the championship match Ryo shot his target in the ear with a .500 Nitro Express bullet (a ludicrously powerful round for hunt against elephants) specially made with a golden tip to get the round stuck in the cranium, and timed it with the champion's punch, making it look that the champion punched him out of the ring and killed him. For added humiliation he waited until the challenger declared he would win in the next round to pull this, and by Ryo's face it was something he was quite used to do. Finally, there's the matter of the payment: Ryo's client was terminally ill of cancer and wanted to pay Ryo with her life insurance, or, in her words, her life, and return drowning herself into alcohol, but Ryo ripped the evidence of the life insurance and asked her to allow him to make her remaining days happy.
In one arc, free lance journalist Yoko Ono had been kidnapped by a delinquent gang to have Ryo walk in their trap, and, thanks to the father of their leader being the boss of the Unryu yakuza gang, were all equipped with guns. Ryo had to save her while avoiding showing his face, as Yoko wanted to make it public and had a camera. So Ryo showed up with fake glasses, nose and moustache (he had arrived with a Princess Peach mask, but discarded it almost immediately), and started advancing in a Slow Walk while shooting the guns out of their hands. But they were many, and Ryo finished his bullets quickly... At which point he extended his hand, grabbed one of the guns he had blasted in air that was landing near him, and resumed fire until he disarmed them all. Then he reloaded.
To complete the moment... The Unryu boss was there, sure of himself and that whoever had beaten up his son multiple times would be defeated but ready to enter the frame if necessary. When Ryo reloaded, he called him... And begged for forgiveness, as he had understood they had pissed off the freakin' City Hunter.
It gets a reprisal a few story arcs later: a yakuza group had kidnapped Kaori and Ryo's latest client, so Ryo barged in their headquarter... And upon seeing him, the Unryu boss started begging for forgiveness again without even knowing why Ryo was there. When Ryo informed him, he immediately ordered their release and revealed who had asked him to kidnap Ryo's client.
The episode where Ryo discovers his fear for airplanes has two moment of pure awesomeness related to the airplanes. The first happens when the villains manage to close him in an airplane. They feel safe, sure they've won... When Ryo stops crying in fear and destroys the plane from inside. Later the villains, defeated, try and run away in airplane using Kaori as hostage... And Ryo instantly conquers his fear long enough to save Kaori.
In his first named appearance, Umibozu's very name caused Ryo an Oh Crap, well after Ryo had already established his credentials as a basically unstoppable One-Man Army. The rest of the story arc, having Umibozu trying to kill Ryo's charge (who had actually hired Umibozu in the first place), was basically a battle of Cool Versus Awesome, having things like the two of them having Casual Danger Dialogue while aiming bazookas at each other. Thankfully Ryo managed to bribe Umi into helping snap their mutual client out of her suicidal intentions...
In another story arc, Umibozu was shot in the back as he protected the daughter of his late commander when he was a mercenary, and, when they told him he needed to go to hospital, he expelled the bullets from his back and closed the wounds. By flexing his muscles. Apparently mere .38 caliber rounds can't harm him...
Same story arc as above: Umibozu successfully disguised himself as a duck.
In Miki's introductory arc he and Ryo had a duel. They shot each other at the same time. Ryo had swapped the bullets with paint ones, but Umibozu managed to match one of the fastest gunslingers in the world!
In the following story arc Umibozu provided the identity of the Villain Of The Week, and when Ryo brought his charge to confront him Umi beat up all of his charges in an Off Screen Moment Of Awesome. Then, in a CMOA mixed to CMOF, juggled with the villain and two of his goons.
An even later story arc gives us an actual Duel to the Death between Ryo and Umibozu in a graveyard. It's pure awesome, and Ryo barely wins (Umibozu survives because Ryo exhausted his ammo by destroying his gun).
This story arc retroactively amped up Umibozu's awesomeness by revealing he was almost blind, and became fully blind during the duel. That's right, Umi is a very skilled crack shot in spite of being near blind. The only reason he didn't go further is that both Ryo and him can feel a foe's killer instinct...
In one occasion, Ryo won a duel with an empty gun. The sweeper known as the Bat was bragging that he could have defeated Ryo if he had a super-accurate gun like him, so Ryo dared him to fight after they exchanged their guns, and when the Bat gave him an empty one Ryo realized the deception and threw it at him, outdrawing the Bat in the process.
In a story arc, Ryo had been hired by a Yakuza group to guard the widow of their late boss, and a rival group burned down a park in their territory in the attempt to both Kick the Dog and convince her to sell the land. When the Yakuza who had hired Ryo arrived to the skyscraper of the rivals to exact revenge for making the widow cry, they discovered that Ryo had already defeated the whole rival group and tied them up to a crane while Kaori blew the building up.
After exterminating three quarters of them, Umibozu and Ryo have to deal with the survivors all at once. Umibozu decides to take on them all to let Ryo go and save Kaori... And after the general's defeat we see that Umibozu has crushed them without being even scratched.
For the rescue... Kreutz had his last ten soldiers surround Kaori and threaten to kill her if Ryo didn't surrender. Ryo came out, threw down his Python... Then declared that, for the woman he loved, he'd survive at all costs... And shoots the bodyguards with two rocket launchers. He had taken the explosive out of the warheads to not harm Kaori, but he still knocks out five of them... At which point he recovers his Python and kills the survivors before knocking out Kreutz with a chop.
Eriko Kitahara's arc shows exactly how Crazy-Prepared and Properly Paranoid Ryo is, and that his Badass Longcoat isn't just owned by a badass but actually badass in its own right: that coat contained a dismounted revolver with ammunition, plastic explosive and condoms, and its sleeves will release tear gas when set on fire. The villains, who had captured him and taken away his Python, were completely caught by surprise and utterly annihilated before he took back his Python. Then they tried to get their revenge by running over him with a bulldozer, that had the mobility advantage on that terrain and was invulnerable to pistol fire... At which point they found the hard way that the coat had a last trick: when shot in the right place, it exploded, destroying the bulldozer.
Made more awesome by Eriko's reaction: to her fashion is Serious Business, and she had previously criticized Ryo's taste in clothing in general and his coat in particular, but after seeing what he continued to produce from it she started admiring it.
In the anime, the duel between Ryo and Geruma, the best hitman of the Lodos Mafia. Geruma was armed with an .44 Auto Mag, a gun that, due her form and weight, wasn't a good choice for a quick draw... And yet Geruma managed to outdraw Ryo. Still, Ryo won. How? Easy: first he shot Geruma's bullet, then, as Geruma was comprehensibly freaking out at Ryo's stunt, shot the hitman in the chest.