Partially hyperbolic surface endomorphisms are a family of not necessarily invertible surface maps which are associated with interesting dynamics. The dynamical behaviour of these maps is less understood than their invertible counterparts, and existing results show that they can exhibit properties not possible in the invertible setting. In this talk, I will discuss recent results regarding the classification of partially hyperbolic surface endomorphisms. We shall see that either the dynamics of such a map is in some sense similar to a linear map, or that the map falls into a special class of interesting examples. This is joint work with Layne Hall.