On the one hand, the spectral theorem claims that the dynamics of hyperbolics systems can be decomposed into finitely many independent and elementary pieces(basic sets). On the other hand, there are systems exhibiting in a "persistent" way infinitely many pieces of dynamics (for instance, sinks); this is the so-called Newhouse phenomenon.

In the context of $C1$-generic dynamics, we discuss some results stating the dichotomy tame vs wild dynamics. Tame systems are those having finitely many elementary pieces of dynamics. Moreover, these systems satisfy some weak form of hyperbolicity and some of the properties of the hyperbolic systems. We also explain how that wild dynamics arises.

The problem of classification of smooth dynamical systems had been a reach source of motivation for beautiful constructions and conjectures for
several decades. The history of these conjectures, as well as the current "state of the art" of the subject will be described. Some of the notions to
be covered are: structural stability, Hadamard-Perron Theorem, invariant manifolds, Morse-Smale systems, Smale horseshoe, Kupka-Smale systems, homoclinic picture, hyperbolic sets, Anosov diffeomorphisms, Axiom A diffeomorphisms, Spectral Decomposition Theorem, homoclinic tangencies, Newhouse phenomena, heterodimensional cycles, Palis' Conjectures.

The purpose of the talk is to provide a very general description; no proofs or technical details will be given.