I will first discuss the pioneering work of Hsiang and Hsiang–Lawson on the construction of minimal sub-manifolds using equivariant geometry, and then some extensions of their theory including new existence and compactness results. As an application, we prove a global bifurcation result that yields arbitrarily many embedded minimal spheres in sufficiently elongated ellipsoids of revolution in any dimension. This is based on joint work in progress with Dario Corona, Fabio Giannoni, and Paolo Piccione.