Dual wielding returns in Diablo III, in expanded fashion. Three of the characters can dual wield various types of weapons, and even the Wizard has a sort of dual wielding. Only the Witch Doctor can't do it, as far as we know. Since the issue of what class weapons are restricted has changed numerous times during development, expect more changes on this issue.
As of November 2010:
The Barbarian is able to dual-wield virtually every one-handed weapon he can equip. (Though there are a lot of one-handed weapons he can't use; such as wands and short staves that the spell-casters use.) The Barbarian can not, however, dual wield two-handed weapons, or use them with a shield as well, as he did Diablo II.
The Monk can also dual wield all one-handed melee weapons. [1]. It was initially said that he could only dual-wield his class specific fist weapons, but that changed.
The Wizard and Witch Doctor can not dual wield weapons, though the Wizard at least has the option of wielding a weapon in one hand and an orb (rather than a shield) in the other. Orbs offer offensive bonuses, almost like a second wand or other caster weapon, rather than the defensive bonuses or blocking that a shield provides.
The Demon Hunter can not dual wield melee weapons, but she can dual wield her class-restricted pistol crossbows. All other bows and crossbows are two-handed, and the Monk and Barbarian can not use any type of ranged weapon.
Dual Wielding Skills and Traits
There are no known skills that can only be used while dual wielding.
There is a trait, Frenzied Attacks that both the Barbarian and the Monk can enable. It provides them with (slightly different) critical hit bonuses while dual wielding.
Blizzard on Dual Wielding
Bashiok spoke about the theory behind dual-wielding, vs. using a sword/shield or a two-handed weapon, in early September, 2010. [2]