The talk will consists of a long historical introduction to  the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces  as well as some current results obtained in collaboration with Corinna Ulcigrai  and Minsung Kim. New developments include a better understanding of the asymptotic of  so-called error term (in non-degenerate regime) and the appearance of new exponents in the  deviation spectrum (in degenerate regime).