In 1977 Bogomolov proved that on surfaces of general type with c_1^2>c_2, curves of a given geometric genus form a bounded family. The role played by foliations in his proof was further investigated by McQuillan, who in 1998 proved the Green-Griffiths conjecture for the same class of surfaces.In this talk I will review some basic properties of foliations on algebraic surfaces, with a focus on birational geometry as initiated by Brunella, McQuillan and others. I will then discuss the problem of their variation in families, and present the main ideas behind the proof of the stable reduction theorem in this context.