I will discuss two different types of essential coexistence of regular (zero Lyapunov exponents and hence, zero entropy) dynamics and chaotic (non-zero Lyapunov exponents) dynamics in the setting of smooth dynamical systems, both with discrete and continuous time. I will review some recent results in this direction, discuss some open problems and describe some new examples of coexistence. In particular, I will outline a construction of a volume preserving topologically transitive diffeomorphism of a compact smooth Riemannian manifold which is ergodic (indeed is Bernoulli) on an open and dense subset of not full measure and which has zero Lyapunov exponent on the complement of this set. I will also discuss a continuous-time version of this example. These constructions demonstrate a "complete" KAM-type picture in the volume preserving category (in both discrete and continuous-time).