Quasiregular maps are higher-dimensional analogs of analytic functions, and non-injective generalizations of quasiconformal maps.  The first part of the talk will focus on the role of distortion and smoothness in the branching behavior of Euclidean quasiregular maps.  I'll sketch the construction (joint with Kaufman and Wu) of $C^{1,\alpha}$ smooth branched quasiregular maps in dimensions four and higher.  The concept of quasiregularity extends naturally to Riemannian manifolds, and a Riemannian $n$-manifold is said to be quasiregularly elliptic if it receives a nonconstant quasiregular mapping from $R^n$.  Recent developments in analysis in metric spaces motivate the study of quasiregularity in non-Riemannian settings, for instance, the sub-Riemannian Heisenberg group.  In the second part of the talk, I'll discuss recent work (joint with F\"assler and Lukyanenko) on Heisenberg quasiregular ellipticity of contact sub-Riemannian $3$-manifolds."