After recalling the main geometric properties of 3-quasi-Sasakian manifolds, I will present our preliminary results on the topology of this class of Riemannian manifolds. They include both 3-cosymplectic and 3-Sasakian manifolds as special cases of minimum and maximum rank, respectively. Recently, there has been an increasing interest toward these manifolds, especially the 3-Sasakian case, due to their relevance in theoretical physics. It is known that any 3-quasi-Sasakian manifold is locally the product of a Hyper-Kaehler manifold and a 3-Sasakian manifold. Nevertheless, we find examples of compact 3-quasi-Sasakian manifolds which are not the global product of a Hyper-Kaehler manifold and a 3-Sasakian manifold.