Limit Breaks have undergone a number of changes when compared to the Limit Break system in Final Fantasy VII. The most important change is that Limit Breaks are only become available to use when a character goes below a certain HP level (specifically, 32%). You will know when they have dropped low enough because that character’s name and HP will be highlighted yellow in the battle screen.
You can also use Aura magic to allow the usage of Limit Breaks without requiring the character to have low HP.
Even if your character has low HP or even if your character has Aura Magic activated you still may not see the option to perform a Limit Break come up on the screen. Thankfully there is a nifty little trick you can use to force a Limit Break. Just keep switching between characters (using the Circle button) until the Limit Break option comes up.
Click on one of the characters/Limit Breaks below for more information on each character’s Limit Break.