Horse Archer

"Wil: "So, the archers of Sacae, they're all mounted, aren't they? I'd have trouble just staying on... If I had to shoot, too, I'd be in trouble!" Rath: "...The beasts of Sacae are swift... If we could not shoot from the saddle, we would starve...""

- FireEmblem: Blazing Sword

Exactly What It Says on the Tin: someone who shoots a bow while on horseback.

Truth in Television for countless civilizations, of course, with many historical armies being made up entirely or primarily of horse archers. The Huns and Mongols are the best-known of these, and have spawned any number of clones in fantasy literature, but almost every civilization that had horses has used these at some point, often to devastating effectiveness, and the Samurai used the dai-kyu (a type of recurved longbow) from horseback up until the Meiji Restoration. Often specialize in Hit and Run Tactics, especially with archers who can spur their horse, turn around and shoot backwards as their enemies pursue. Naturally, since it requires expert horsemanship, a specialty of any Born in the Saddle culture. An honorable mention goes to earlier civilizations (Persians, Egyptians, etc.) that used chariots as a shooting platform, but in those cases it was a team of both horses and men.

Even in the era of the gun, firearms were almost impossible to use effectively from horseback, as a muzzle-loaded weapon requires stability, dexterity, and two hands to reload. This meant that bow-wielding horsemen held on to an advantage up until the invention of the revolver.

The MythBusters also proved that charging towards your target on a horse while firing can impart your arrows with ~70% more kinetic energy, and increase their penetration power a bit.

In Tabletop Games and Video Games, these are often Game Breaker units, but not always. For balance reasons, they are often less powerful than archers who go on foot, and if Annoying Arrows are in place already, this can lead to them being unable to effectively damage the enemy. Given that they often trade armor for speed, they can also easily be portrayed as Fragile Speedsters. If they can attack with swords as well, it's a case of Bow and Sword in Accord.

Film

 * Badass Princess Mérida from Brave is an ace with a bow and arrow, and she can hit a perfect bullseye while on horseback.

Literature

 * The Dothraki and the Dornishmen from A Song of Ice and Fire.
 * The Mongols in all books, naturally enough.
 * The various steppe tribes in Harry Turtledove's Videssos books. The Videssians themselves may also qualify, though they prefer more armor than the average horse archer.
 * In The Lord of the Rings the Rohirrim field a number of these. However most of their army seems to be lancers.
 * The Haldane Household Archers function this way in battle, as seen in The King's Justice. In that same book, Kelson himself is one.
 * In Belisarius Series the Persians prefer steppe style bows that have a great rate of fire. Romans prefer giant bows that can penetrate armor. Rajputs are also notable Horse Archer s but not as good as Persians.
 * In Harald, The Westkin tribes are the lightly armored sort, and the Vales have cataphracts who wear heavier armor.
 * In the Heralds of Valdemar series, the Skybolt mercenary company is primarily composed of horse archers trained in skirmish tactics. The rest of the company are dirty tricks specialists.
 * Implied to exist in the Zlobenian army in Monstrous Regiment, since the Borogravians get hold of a powerful crossbow called a "horse-bow".

Live-Action TV

 * In the Power Rangers Samurai TV movie "Clash of the Red Rangers", Blue Ranger Kevin uses his Hydro Bow while on horseback.

Video Games

 * Mount & Blade has the Khergit faction, almost every single one of their faction troops is one of these (with the exception of heavier lancers in the Warband expansion). It's also possible to make your player character one.
 * If you choose to start in the Khergit Khanate and you don't put any points into riding or horse archery and only use infantry in your battles... Well even the small packs of roaming steppe bandits that provide low level characters exp will stomp all over you. In fact the steppes is probably the hardest starting region because of the prevalence of horse archers.
 * Age of Empires has them, along with several varieties such as Chariot Archers and Elephant Archers. II has Horse Archers, and two civilisations have them as unique units, namely the Mongol's Mangudai and the Spanish Conquistador (which might not count since it uses a gun, not a bow). III has Dragoons which are basically the same as the Conquistadors from the previous game, some civilisations who have actual horse archers instead of the more usual Dragoon, and the Indians even have gunners firing from camelback. The spin-off, Age of Mythology has the Turma which throws javelins from horseback, the Centaur, and the Chariot Archer from the original game makes a return.
 * Shining Force I and II have Lyle and May, centaur archers who are one of the best ranged characters in their respective games.
 * The Total War series has them as a staple unit for most cavalry centered factions particularly in...
 * Medieval Total War I and II finds them among most of the non-Catholic factions.
 * Rome Total War also have them as the core unit of several eastern/steppe factions and Rome itself in the Barbarian Invasion expansion.
 * Fire Emblem has had a couple of units across the series that fill this role; additionally, this trope is regularly discussed by units of said trope in games which have support conversations. The most famous are:
 * Midayle, Lester, Andorey, Scorpio, promoted!Leaf and promoted!Lachesis. Vylon is supposed to be this due to being a Master Knight, but he doesn't have a bow (FE 4);
 * Selphina (FE 5);
 * Sue, Dayan and Shin (FE 6);
 * Rath and Uhai (FE 7);
 * King Hayden, Ranger!Gerik and Ranger!Neimi (FE 8);
 * Astrid (FE 9 and FE 10).
 * Warhammer Fantasy Battle has Kislevite Horse Archers, Hobgoblin Wolf Riders, Wood Elf Glade Riders, and Empire Pistoliers (who, as the name suggests, aren't quite horse archers, but definitely ride horses).
 * The Night Elves in Warcraft III can promote their normal archers to Hippogryph riders, making them both mounted and flying.
 * The Hero Unit Priestess of the Moon is this with a tiger as the mount.
 * Wander In Shadow of the Colossus can do this while riding Agro.
 * Link can do this while riding Epona in some games in the The Legend of Zelda series.
 * A basic mounted unit in Civilization IV. Appears as unique Mongol unit in Civilization V, while the Indians get Elephant Archers.
 * A patch in Skyrim allowed the player to become this, although it is a bit hard since you have to aim without a retical.
 * Master of Magic has Horsebowmen, as a Nomad unit with unique type. But there are many other non-ship units that can move and shoot. Centaurs (Beastmen) are very close: slightly better defenses, but weaker melee attack. Pegasai (High Elf) are flying (though expensive and low figure count makes them less effective despite great stats). Since most Dark Elf units have Ranged Magic attack, their Cavalry is close enough too, and Nightmares (elite unit) are more powerful counterpart of Pegasai.
 * Of course, many non-cavalry units have similar functionality, Rangers (Nomad) and High Elf Magicians just walk that fast, etc. And since Endurance spell increases Movement, it turns any shooty unit into move-and-shoot unit.

Web Comics

 * Girl Genius Geisterdamen use crossbows while riding their spiders. Not very accurate when both shooters and their target are bobbing, but still a threat.

Web Original

 * Featured prominently in the Tales of Ubernorden story The Sword of War.

Real Life

 * The Mongols used this tactic to take over most of Asia.
 * The Byzantine cavalry, based on a careful balance of lancers and archers.
 * Yabusame (流鏑馬) is the Japanese traditional sport of mounted archery.
 * Parthian horse archers were devastating against the Romans at Carrhae (53 BCE)-- particularly impressive in that they did so without stirrups. (Incidentally, it's often claimed that the phrase "parting shot" is a corruption of "Parthian shot". In truth it's a coincidence and the two were coined separately.)
 * The Parthians in turn learned it from the Scythians, who successfully prevented the armies of both Darius and Alexander the Great from expanding their empires into the Northern Ukraine and Russia through evasion and harassing.
 * The Comanches in North America managed to stall the expansion of the US. Impressive with the difference in technology level.
 * Medieval Hungarians.
 * Medieval Swedish and German mounted crossbowmen. Their fire rate was slower than that of the mounted archers, but they used far more powerful bows.