In this talk, we will discuss foliations and their transverse invariant measures (i.e., measures on cross-sections that are invariant under the holonomy maps) from a dynamical systems point of view. We will show that for a large family of diffeomorphisms, the unstable foliations admit families of transverse measures that are naturally related to certain probability measures invariant under the dynamics. Given an unstable leaf, we will consider a dynamically defined average that captures its intersection with cross-sections and prove that this averaging will converge exponentially fast to the transverse invariant measures. This is a joint work with Ures, Viana and J. Yang.