  We survey some recent advances in the study of (area-preserving) flows on
surfaces, in particular on the typical dynamical, ergodic and spectral
properties of smooth area-preserving (or locally Hamiltonian) flows, as well as
recent breakthroughs on linearization and rigidity questions in higher genus.
We focus in particular on the Diophantine-like conditions which are required to
prove such results, which can be thought of as a generalization of arithmetic
conditions for flows on tori and circle diffeomorphisms. We will explain how
these conditions on higher genus flows and their Poincare' sections (namely
generalized interval exchange maps) can be imposed by controlling a
renormalization dynamics, but are of more subtle nature than in genus one since
they often exploit features which originate from the non-uniform hyperbolicity
of the renormalization.
