One of the major results in the study of 3-manifolds is the fact that most 3-manifolds have a finite cover that fibers over S1. One may ask what is the counterpart of this result for other classes of manifolds. In this talk we will discuss the case of smooth projective varieties (or more generally Kaehler manifolds) and present some geometric and group-theoretic aspects of  "virtual algebraic fibrations" of their fundamental groups.