The talk will survey progress over the past several years (most recent of which joint work with Costante Bellettini) on the question of regularity and compactness for a large class of uniformly area bounded, mean curvature controlled hypersurfaces (codimension 1 integral varifolds) of a Riemannian manifold. Subject to the requirements that the orientable embedded parts of the hypersurfaces have mean curvature prescribed by an ambient \(C^{1, \alpha}\) function \(g\) and have Morse index (with respect to the relevant functional determined by \(g\) ) bounded uniformly, and additionally that the hypersurfaces satisfy two necessary structural conditions, the work provides a sharp size upper bound for the singular set \(S\) which says that \(S\) has codimension at least 7. In the case that \(g\) is a constant and the hypersurface is weakly stable, curvature estimates (joint work with Costante Bellettini and Otis Chodosh) follow a posteriori. The theory to be discussed builds on, generalises and unifies the pioneering work in regularity theory of many including De Giorgi, Federer, Almgren, Simons (on locally area minimizing currents), and Schoen-Simon-Yau and Schoen-Simon (on stable hypersurfaces with small singular sets) dating back to the period 1960-1980.