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.