Constant mean curvature (CMC) hypersurfaces arise naturally as critical points of the area functional under a volume constraint. Stability means that the hypersurface locally minimizes the area functional under small variations that preserve the enclosed volume.

In this talk, we will revisit classical results and share recent insights into the stability of CMC hypersurfaces.

The discussion will draw on techniques from PDE theory and differential geometry  and some open problems will also be highlighted.