We survey recent developments in the area of geometric function theory and nonlinear analysis and in particular those that pertain to recent developments linking these areas to dynamics and rigidity theory in dimension n>2. A self mapping (endomorphism) of an n-manifold is rational or uniformly quasiregular if it preserves some bounded measurable conformal structure. Because of Rickman's version of Montel's theorem there is a close analogy between the dynamics of rational endomorphisms of closed manifolds and the classical Fatou-Julia theory of iteration of rational mappings of the Riemannsphere. The theory is particularly interesting on the Riemann n-sphere where many classical results find their analogue, some of which we discuss here. We present the most recent results toward a solution of the Lichnerowicz problem of classifying those manifolds admitting rational endomorphisms. As a by product we discover interesting new rigidity theorems for open self maps of closed n-manifolds whose fundamental group is word hyperbolic.