In this talk I will introduce the notion of CR submanifolds of a Sasakian manifold, which is close to the classical concept of CR submanifold of a complex manifold, and it generalizes many definitions already known in literature. As consequence of the fact that CR submanifolds are naturally endowed with a CR structure, I will discuss two main global results. The first one is a Frankel type theorem which provides sufficient conditions for two CR submanifolds to have non empty intersection, while the second is a non existence result for Einstein hypersurfaces for a class of regular Sasakian manifolds. The talk is based on joint works with Antonio Lotta (Bari).