Abstract: We will discuss a variety of weak and strong compactness results for sequences of minimal hypersurfaces (with and without boundary) in compact Riemannian manifolds of dimension between 3 and 7. In particular, exploiting a precise bubbling analysis, we obtain new strong compactness theorems in dimension 3 that extend results of Choi-Schoen and Fraser-Li from ambient manifolds with positive Ricci curvature to manifolds with only positive scalar curvature under the additional assumption of an index bound. The presented results have been obtained in joint work with Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp.