  In this paper, we study the translation surfaces corresponding to meromorphic
differentials on compact Riemann surfaces. We compute the number of connected
components of the corresponding strata of the moduli space. We show that in
genus greater than or equal to two, one has up to three components with a
similar description as the one of Kontsevich and Zorich for the moduli space of
Abelian differentials. In genus one, one can obtain an arbitrarily large number
of connected components that are easily distinghished by a simple topological
invariant.
