We consider area-preserving flows on surfaces which are locally given by smooth Hamiltonians. It turns out that the presence or absence of mixing depends on the type of fixed points. We proved in our PhD thesis that the presence of centers is generically enough to create mixing. Recently we showed that if such flows have only saddles, they are generically not mixing, but weakly mixing. The results use the flows representation as suspensions over interval exchange transformations and the study of deviations of Birkhoff averages over interval exchanges.