An extension of a result of Sela shows that if $\G$ is a torsion-free word hyperbolic group, then the only homomorphisms $\G\to\G$ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if $M$ is a closed Riemannian $n$--manifold with negative sectional curvature ($n\neq 4$), then every quasiregular mapping $f:M \to M$ is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that $f$ is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open automorphism. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no $\pi_1$-injective proper quasiregular mappings $f:M\to N$ between hyperbolic $3$-manifolds $M$ and $N$ with non-elementary fundamental group.