Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Consider the moduli space of pairs (C,) where C is a smooth compact complex curve of a given genus and is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.