Quasiconformal and quasiregular maps are a generalization of conformal and holomorphic maps, with a rich theory even in dimensions higher than two. In the same vein, uniformly quasiregular maps aim to generalize aspects of holomorphic dynamics to higher dimensions. In two dimensions, the only compact Riemann surfaces which admit non-injective holomorphic dynamics are the sphere and the torus, and in the case of the torus the dynamics is locally injective. In this talk, I'll discuss the corresponding classification question for uniformly quasiregular maps in higher dimensions. In particular, an approach from joint work with Pekka Pankka has resulted in new obstructions, such as that a closed manifold admitting a non-trivial uniformly quasiregular map has uniformly bounded de Rham cohomology.