Wyvern

A  is a flying, non-solid enemy, added and only available in the 1.1 update and after. It slightly resembles the face of a dog and the body of a serpent/dragon with hands and feet.

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It has a total of 4,000 health, and will only spawn after the player has defeated the Wall of Flesh and when the player is at sky level (300+ feet, 500+ feet, and 1,000+ feet on small, medium, and large worlds respectively). Wyverns spawn at the same altitude as Harpies. It seems to spawn lower than this on Blood Moons and in The Hallow. This enemy is a mini-boss, and can be dangerous for players without good armor and defense, as the head can hit an undefended player for up to 90 damage, while the body delivers about 40. This is still dangerous because the Wyvern tends to charge into you repeatedly, and lingering in one spot while it is hitting you can rack up a lot of damage. Multiple Wyverns can spawn in one world.

Strategy
Obviously, if you are really good at using Gravitation Potions and have good enough weapons you don't really need to use these strategies.

Wood Platforms
A good strategy is to make 2 wide platforms out of Wood Platforms, one above the other, and when the Wyvern appears, stand on the lower platform and just move left and right a little. Time your jump to the mowwrwrr3r3r3rr3rr33rrr3r3r3r3r3ment the head almost touches you. You should dtertgetergergergdfgrgodge its attacks most of the time.

Skybridge
Make a Skybridge in the atmosphere (where gravity has less effect.) Use at least ~300 blocks so you can do a full run, jump, and float with Spectre Boots + Cloud in a Balloon twice. Build either Cobalt Repeater or Mythril Repeater, and use Jester's Arrows. Cobalt Armor with magic buff and craft Crystal Storm. Then run fast so the Wyvern follows you in a more or less straight line, shooting him with Crystal Storm along his length until you're out of mana, then switch to the repeater. Once your mana is full again, switch back. If you reach the end of your bridge, turn around and run the other way. Using the atmosphere this way makes it easier to dodge the Wyvern's attacks.

Flight Items
With Rocket Boots, Spectre Boots, or Wings, you can go through a repetitive motion of jumping and flying to the right, attack from above, wait for him to come back into sight, rinse, repeat. By doing this it will continue to fly in a predictable pattern, making dodging its attacks easier. Grenades works very well since all the segments of its body can take damage. A grenade will be able to hit up to 6 segments of its body because of the explosion, but usually hits about 4. The Flamethrower is also a good option, though obtaining it requires defeating a Hard Mode boss.

Flails
Get a relatively powerful flail, (Sunfury or a Dao of Pow). Once a Wyvern spawns, use a Gravitation Potion to alternate falling up and down while staying horizontally still. As the Wyvern is approaching you from above or below, aim directly along the length of its body with your weapon of choice. With a Dao of Pow, you can take at least 1000 health off of this creature per swing.

Piercing Magic
(Note: this strategy works better with Rocket Boots and a Cloud in a Bottle, or other jumping items.) Buy the Ice Rod and cast it under you while jumping up. Get the Wyvern's attention and jump back down. As you fall, aim up at it with a Magical Harp or Demon Scythe. Depending on which weapon used, the Wyvern will be killed outright or will have 1000/4000 HP left. Doing this on a large map is ideal, as even the Demon Scythe will be able to kill it before you reach the ground.

Nurse
Build a house for the Nurse away from other NPC's. Summon the Wvyern, and stand right above where the Nurse is, making sure you can still talk to her. The Wvyern will do a figure of 8 to hit you (when you are standing still), so use something like the Dao of Pow, and just hold it, healing with the nurse whenever your health is low.

Freefall with Magic Mirror
Build a small house at the top of your world with a bed inside. Set this house as your Spawn Point. Now, simply jump out and begin attacking the Wyvern with your weapon(s) of choice. If the Wyvern doesn't die, just teleport back to the house and start over.

Cursed Flames
Enter a house on a Floating Island and close the door (alternatively, you can just build a house high in the sky). Wait until the Wyvern spawns, and wait until the Wyvern "sticks its head in" and then use Cursed Flames until you run out of mana. If done correctly, the flames can bounce several times inside the room and hit the Wyvern multiple times, dealing up to 2000 damage. Then, you can proceed to finish it with any method above.

Spiky Balls
Proceed as above to a house on a Floating Island. When you are inside, rather than waiting for the Wyvern to spawn, immediately cover the floor with Spiky Balls. This way, the Wyvern will take a fair amount of damage every time it swoops down to "ground" level to attack you. Keep in mind that Spiky Balls disappear soon after being thrown and are strictly ground-based weapons. As such, you may want to prepare one of the other strategies above as a backup in case you start to take too much damage while in the house.

Trivia

 * The Wyvern looks very similar to Haku (in his dragon form) from the movie Spirited Away. Some believe it also bears a vague resemblance to Falkor the Luck Dragon from The Neverending Story, as well as the White Dragon from Final Fantasy 4, which makes sense because it was added to Terraria at the same time as a Final Fantasy Based Costume (Clown Clothes) and Weapon (Breaker Blade).

General Tips

 * Using the Dao of Pow can be extremely effective, however like any burrowing enemy, it cannot be confused.
 * Using Ranged weapons can be much more effective than melee. Good weapons would be the Starfury, the Starcannon, the Shotgun, any Repeater, the Megashark, or the Clockwork Assault Rifle.
 * If you have a Demon Scythe, Water Bolt, Laser Rifle or Magical Harp, it may take very high damage if lined-up with the projectiles' trajectory.
 * For first timers, it would be wise to go against this enemy in singleplayer to get used to its pattern.
 * A Wyvern is able to follow you to your home as it can fly faster than you even if you're wearing Hermes Boots. Do not run away and go back to your home as it may catch you by surprise.
 * They will still continue to follow you back to your home even if you teleport using a Magic Mirror.
 * The Wyvern has a long body, which can be turned to your advantage by building a small room and using attacks that ricochet off walls (Cursed Flames, Crystal Storm, Light Disc, etc.) inside. As the Wyvern charges through, it will be hit by each projectile multiple times, increasing your damage potential.
 * If you fight the Wyvern in outer space, it is extremely easy to dodge every single one of their attacks with decent timing. The only downside is the increased Harpy spawn, but they are little more than an annoyance.

Bugs

 * Sometimes, if a Wyvern is killed in multiplayer when it's laggy, its head will stay and keep attacking you.
 * Wyverns may get knocked back a little bit when hit by an attack in laggy multiplayer servers.
 * If you die while fighting the Wyvern, it may follow you back to your spawn point.