It is well-known that the dynamics of uniformly hyperbolic systems decomposes into finitely many transitive pieces (the basic sets). A natural generalization for the decomposition of non-uniformly hyperbolic systems are the homoclinic classes.

I will present recent results that allows to describe the dynamics of arbitrary surface diffeomorphisms, up to neglect sets having zero entropy: in particular, I will discuss the construction of symbolic codings of each class and the properties of the equilibrium measures.