We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measure-theoretic entropies. These new invariants are only non-zero for non-invertible maps, and they give a quantitative measurement of how far a given map is from being invertible.We obtain analogs of many known results for topological and measure-theoretic entropies.In particular, we obtain product rules, power rules, analogs of the Shannon–Breiman–McMillan theorem, and a variational principle.