Each class has two level 60 sets limited to them - one occupying the primary and secondary hands, and the other occupying 5 item slots (or 6 for the Barbarian's Immortal King set). Below is the list of them all. Notice that the larger sets only require 4 items (or 5 for the Immortal King) to get the full set bonuses.

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Post-1.0.4 class-specific sets:

Barbarian:

• Bul-Kathos's Glory: 1-Handed Mighty Weapon, 1-Handed Mighty Weapon
• Immortal King's Legend: Mighty Belt, Gloves, Boots, Chest Armor, 2-Handed Mighty Weapon, Helm

Demon Hunter:

• Danetta's Creed: Hand Crossbow, Hand Crossbow
• Natalya's Solace: Helm, Ring, Hand Crossbow, Cloak, Boots

Monk:

• Shenlong's Noble Path: Fist Weapon, Fist Weapon
• Inna's Grandeur: Belt, Pants, Daibo, Chest Armor, Spirit Stone

Witch Doctor:

• Manajuma's Ornaments: Ceremonial Knife, Mojo
• Zunimassa's Whispers: Voodoo Mask, Mojo, Boots, Ring, Chest Armor

Wizard:

• Chantodo's Return: Source, Wand
• Tal Rasha's Sacrifice: Amulet, Belt, Chest Armor, Helm, Source

And because they are still in circulation,

Pre-1.0.4 class-specific sets:

Barbarian:

• Bul-Kathos's Children: 1-Handed Mighty Weapon, 1-Handed Mighty Weapon
• Immortal King: Mighty Belt, Gloves, Boots, Chest Armor, 2-Handed Mighty Weapon, Helm

Demon Hunter:

• Danetta's Oath: Hand Crossbow, Hand Crossbow
• Natalya's Wrath: Helm, Ring, Hand Crossbow, Cloak, Boots

Monk:

• Shenlong's Defense: Fist Weapon, Fist Weapon
• Inna's Majesty: Belt, Pants, Daibo, Chest Armor, Spirit Stone

Witch Doctor:

• Manajuma's Ritual: Ceremonial Knife, Mojo
• Zunimassa's Spirit: Voodoo Mask, Mojo, Boots, Ring, Chest Armor

Wizard:

• Chantodo's Legacy: Source, Wand
• Tal Rasha's Wrappings: Amulet, Belt, Chest Armor, Helm, Source