Abstract : Interesting examples of group actions on measure spaces are given by actions on homogeneous spaces. The setting is as follows. Let G be a locally compact group (often a Lie group or a linear algebraic group) and H a lattice in G. Every group of affine transformations of G preserving H acts on the homogeneous space G/H in a measure preserving way. We will discuss some questions concerning these actions, such as ergodicity, existence of a spectral gap or strong ergodicity (in the sense of K. Schmidt). The results we will present concern two classes of groups, requiring different approaches: the case where G is semisimple and the acting group a consists of translations and the case where G is nilpotent and the acting group consists of automorphisms.