Abstract : Given a nontrivial conjugacy class $g$ in a free group $F_{N}$, what can we say about the typical growth of g under application of a random product of auto-morphisms of $F_{N}$? I will present a law of large numbers, a central limit theorem and a spectral theorem in this context. Similar results also hold for the growth of simple closed curves on a closed hyperbolic surface, under application of a random product of mapping classes of the surface. This is partly joint work with François Dahmani.