The classification of horocycle invariant measures on finite volume hyperbolic surfaces with negative curvature is known since the work of Furstenberg and Dani in the seventies: they are either the Haar measure or are supported on periodic orbits. This problem fits inside the more general problem of the classification of horospherical measures in finite volume homogenous spaces.

In this talk, I will explain how similar questions arise in the moduli space of abelian differentials (and more generally in any affine invariant manifolds) and will discuss a notion of horospherical measures in that context. I will then report on progress toward a classification of those horospherical measures and related topological results. This is a joint work with J. Smillie, P. Smillie and B. Weiss.