Wyvern

A  is an enemy added in the 1.1 update and after. Its face slightly resembles that of a dog's, and it has the body of a serpent/dragon with hands and feet. It is the weaker version of the Arch Wyvern (console only).

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 very strong, 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, even with good Armor.

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

The Wyvern seems to have a certain attack pattern, every time it goes up it does a quick swirl and then comes straight down towards you. If you jump when it goes up the Wyvern will just touch you.But at that point if you move to the side it'll miss you completely which then you are able to attack it.

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 moment the head almost touches you. You should dodge its attacks most of the time.

Skybridge (with video)
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. 350px

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. The Demon Scythe also works well. Jump when the Wyvern is approaching from above, and, if timed right, the head and several pieces of the body will pass through it. The same concept applies for throwing weapons such as the Thorn Chakram or the Flamarang. The Wyvern will run along the edge of the weapon, taking multiple hits before it bounces off. The speed of the weapon will also allow for multiple hits in one run. A Golden Shower can also be good for killing it, since it can lower his Defense since the Golden Shower inflicts Ichor and does a decent amount of damage. It also uses very little Mana, and has fast speed. And it is a pretty easy item to get, considering you need to unlock Hardcore Mode. A Clockwork Assault Rifle could also be useful for this boss, due to its speed and strength. This item is obtained from the Wall of Flesh. Ichor Bullets would also be useful once again, because it lowers his defense, and you may be able to do more damage than with the gold shower.

Swords
If you have a relatively powerful sword, like an Ice Sickle or one of the higher tier ore swords (Fiery Greatsword or better), you can kill him without taking damage as his attacks can be easily dodged with good timing. He always charges from above you coming into close range which is ideal for swords usage. You should have Hermes Boots or better as they can make the difference in dodging his attacks. When the Wyvern comes down at you from the left side, you should run left to get just past his head below him. If you don't run too far so he barely misses you, he usually goes straight down not turning around to hit you and when he passes you, turn away to the other side. When the Wyvern comes from the right side, you should run right to get just past his head below him. As he goes straight down passing you, you can do a load of damage with your melee weapon hitting multiple segments. This doesn't require some sort of arena. The running space a floating island gives usually is more than enough to dodge the attacks. This way is very useful when you get attacked by a Wyvern while exploring floating islands, when there is no time to prepare.

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.

If you have the Nimbus Rod, you can get high enough and place a couple of clouds, and the Wyvern will continue to fly through the damaging rain. Better yet, get high enough and as you fall, the Wyvern will follow you straight down and be constantly hit by the rain, killing him within just a few moments.

Line up some Golden Shower streams on him and dodge him with rocket boots and/or cloud in a bottle/balloon and the fight will be a walk in the park.

Nurse
Build a house for the Nurse away from other NPC's. Summon the Wyvern, and stand right above where the Nurse is, making sure you can still talk to her. The Wyvern 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.

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, especially those which could pierce through enemies (Flamethrower), or have a really high DPS. (Megashark)
 * 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 single player to get used to its pattern.
 * A Wyvern moves very quickly, but a player with a Hoverboard or running with Lightning Boots on Asphalt Blocks could outrun them. It is not advised to run back home (When there is NPCs) when you are too slow, as NPCs might get killed while the wyvern is chasing the player.
 * They will still continue to follow you back to your home even if you teleport near them using a Magic Mirror, otherwise, they despawn.
 * 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.
 * Another good way to hurt the Wyvern without getting hit is when he dashes at your, back away from him while flying as he follows you, while doing this use a ranged weapon to shoot his head as he follows you. You will be able to do more damage doing this method since the head only does 10 defense, unlike the body which does 20.  Repeat this method to kill him.

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 Eidolons 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).
 * Strange enough, the text book definition of the Wyvern from legend describes it as a scaly serpentine dragon with only two legs, a sharp or barbed tail and most times, has one pair of wings. This description is completely different than the Terrarian take on the Wyvern which may be due to the fact  the Terrarian Wyvern is a reference to something else, as stated above.
 * The Angler says that Wyverns are born as tadpoles, and refers to them as being "actually like.. well, a frog!"

Bugs

 * Sometimes, if a Wyvern is killed in multiplayer when the server is laggy, its head will stay and keep attacking you. The head will also still do damage, but is killed in one hit of any weapon.
 * 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.
 * A common glitch is that the body will disappear, leaving the head. On multiplayer, it is possible that only one serve has glitched, and the other players are still able to see it. When this happens, the Wyvern can still take and give damage, as the body has not actually 'disappeared', but rather it is 'invisble'.