I will present some methods for proving ergodicity of skew product extensions of interval exchange transformations (IET) of periodic type. I will also deal with a class of smooth flows on noncompact manifolds which are extensions of so called multivalued Hamiltionian flows on compact surfaces of higher genus. These flows have Poincare sections for which the first recurrence map is isomorphic to a skew product of an IET and a BV cocycle or a cocycle with logarithmic singularities. This allows us to prove a sufficient condition for ergodicity whenever the IET has periodic type. The talk is based on two papers joint with J.-P. Conze and C. Ulcigrai.