We introduce a class of branched covers on oriented closed connected topological n-manifolds called expanding combinatorial Thurston maps, which generalize expanding Thurston maps on 2-spheres. We show that, for n ⩾ 3, a visual metric for an expanding combinatorial Thurston map on an oriented closed connected Riemannian n-manifold is quasisymmetrically equivalent to the Riemannian metric if and only if the map is uniformly quasiregular. A fortiori, such an expanding combinatorial Thurston map is uniformly quasiregular if and only if it is a Lattes map. This is based on joint work with Pekka Pankka and Hanyun Zheng.