Let $S$ be a compact (hyperbolic) surface. The space of geodesic currents $C(S)$ can be viewed as the completion of the set of all weighted closed geodesics on $S$, the same way as the space of measured laminations $ML(S)$ is the completion of all weighted simple closed geodesics. In particular, $ML(S)$ can naturally be identified with a subset of $C(S)$. In this talk we look at mapping class group invariant ergodic measures on $C(S)$ and extend the Lindenstrauss-Mirzakhani and Hamenstädt classification of such measures on $ML(S)$ to the space of currents. Essentially, any such measure not supported on $ML(S)$ must be atomic. This is joint work with Gabrielle Mondello.