Kontsevich and Zorich famously classified the connected components of strata of translation surfaces over moduli space. The corresponding problem over the Teichmüller space requires the analysis of which mapping classes can be realized as deformations lying inside the stratum. In this talk, I will present joint work with Nick Salter in which we classify the (non-hyperelliptic) connected components of strata over Teichmüller space. Unlike the case for strata over moduli space, we find that there can be many (but finitely many) connected components, depending on both genus and cone angle.