  We determine the maximum number of induced cycles that can be contained in a
graph on $n\ge n_0$ vertices, and show that there is a unique graph that
achieves this maximum. This answers a question of Tuza. We also determine the
maximum number of odd or even cycles that can be contained in a graph on $n\ge
n_0$ vertices and characterise the extremal graphs. This resolves a conjecture
of Chv\'atal and Tuza from 1988.
