Gromov's compactness criterion implies that the family of closed Riemannian manifolds with dimension bounded above, diameter bounded above, and sectional curvature (or Ricci curvature) bounded below, is pre-compact with respect to the Hausdorff topology in the space of compact metric spaces. The general behavior of a sequence $X_i$ in one of those families is very different depending on whether $vol(X_i)$ goes to zero or is bounded below by a positive constant. In this talk I will present some topological obstructions, involving the fundamental groups of the spaces $X_i$, for the second situation to occur. The main tools used in this kind of result are systolic inequalities, and the Yamaguchi--Burago--Gromov--Perelman fibration theorem in the case of lower sectional curvature bounds.