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Evade is a duck-and-roll maneuver possible in the Diablo 3 console version using the d-pad. It is not a move that can be utilized in the PC/Mac version of the game, and is one of Blizzard's changes to the console version to give it a more "action-y" feeling.
Evade is mapped to the right stick on the Xbox 360 controller, and can be used by simply pushing that stick in the desired direction. The character will immediately dive to the ground and roll over in the selected direction, moving about 10 yards in the process. There is no resource cost or cooldown time for Evade; it can be done repeatedly without delay.
Characters are not immune to all damage types while Evading, but they will usually be missed by monster swings, and as Evade deals 1 point of damage it can be used to break barrels or other destructibles. The principle purpose of Evade is to reposition your character, without changing the direction they are facing.
Evade is helpful since the console controls work without a mouse pointer. Characters on the console automatically face in whatever direction the left stick points them, and they move semi-realistically, taking a moment to turn around and change direction. Evade can thus used, for example, to move (by Evading) right or backwards while still facing to the left. Ranged attackers can use Evade repeatedly to quickly backtrack or dodge incoming projectiles without changing their angle of attack, and any character can move back out of the reach of enemy attacks.
Evade can be used in any direction; simply press the right stick in the desired direction and your character will roll that way. If the direction of Evade is forward or to the side the character does not hesitate to dive into it. Evading backwards (from the direction your character is facing) looks realistic, as the character paused for a slight repositioning of the feet before rolling backwards or backwards diagonally.
Most PC players spend a minute just rolling around when they first try out the console, enjoying the realistic animations and novelty of the ability.