The authors develop a theory of quasiregular maps in a sub-Riemannian geometry of two-step Carnot groups. An analytic definition for quasiregularity is suggested and it is shown that conconstant quasiregular maps are open and discrete maps on Carnot groups which are two-step nilpotent and of Heisenberg type. Some results which are known to be valid in \(\mathbb{R}^n\) are extended to this setting of Carnot groups. In particular, it is established that the branch set of a nonconstant quasiregular map has Haar measure zero and quasiregular maps are almost everywhere differentiable in the sense of Pansu. The authors approach is that of nonlinear potential theory: the Harnack inequality and Hölder continuity of solutions are obtained.