From Diablo Wiki
Increased Attack Speed or "AS" is one of the most sought after properties in Diablo 3, and is one of the trifecta of the most desired offensive stats.
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 Increased Attack Speed
Increased Attack Speed can occur on items or via skills, and helps greatly in damage and survival. There is no cap to Attack Speed and most players try to obtain as much of it as possible, for functional and aesthetic reasons.
Attack speed can be found on magical or rare items (or Legendaries via the random affix) of these item types:
- Off-hand weapons.
Attack Speed is also provided by numerous Legendary and Set items of these item types. It's also found as an inherent bonus on various other item slot legendaries, including some of the most valuable in the game such as Mempo of Twilight (db), Lacuni Prowlers (db), The Witching Hour (db) and many others.
 Benefits of Increased Attack Speed
Damage is boosted by attacks hitting more frequently, but this property provides numerous other benefits, both obvious and subtle.
- Faster attacks are just plain fun, to see your character hitting or casting more rapidly. They can also make a huge difference, such as boosting your character's attack speed enough to get off two shots rather than just one, before a monster can reach you.
- Skills and items that deal deal DoT check that damage at a frequency based on a character's attack speed, thus dealing damage faster with more Attack Speed.
- Various skills for the Demon Hunter, Monk, and Barbarian generate X resource per use, so attacking faster helps these build more resource more quickly.
- Skills and items that debuff enemies or that provide protection or Life on Hit naturally work better with more hits in a shorter time.
- Many effects and procs click per hit or per critical hit, and launching more attacks more quickly will create more of these effects.
On the whole Attack Speed is one of the most valuable modifiers in the game, ranked right there with Critical hit Chance as the most sought after on items.