There are 7 Campaigns and each has a specific criteria to unlock.

Heroes Assemble - Available from the very start of the game. Restricted to HERO characters only.

Villains United - Unlocks upon reaching Player Level 13 and when you have 3 VILLAIN characters. Restricted to VILLAIN characters only.

The Nexus - Unlocks after completing Villains Chapter 3. Open to ALL characters.

Cosmic Conflicts - Unlocks after completing Heroes Assemble Chapter 6 and when you have 5 COSMIC characters. Restricted to COSMIC characters only.

Mystic Forces Rising - Requires player level 55 to unlock. Restricted to MYSTIC characters only.

Doom War - Unlocks after completing the Nexus Chapter 8.

Isotope-8 - Unlocks upon reaching Player Level 55.