The Anosov-Smale uniformly hyperbolic diffeomorphisms form an open set of chaotic dynamical systems well-understood since the 1970s. Newhouse quickly discovered the set of such diffeomorphisms is not dense, prompting the search for larger but tractable classes. Several groundbreaking works by Pesin, Katok, Newhouse, and others led to the idea that all smooth surface diffeomorphisms should behave much like uniformly hyperbolic ones provided one focused on the invariant measures with positive entropy. In our works, we have proved a conjecture by Newhouse (finite number of ergodic measures maximizing entropy) and generalized much of the uniform theory including a spectral gap property and Viana's conjecture (existence of a physical measure if there are positive Lyapunov exponents on a set of positive volume) for all smooth diffeomorphisms on surfaces.