I will discuss the pioneering work of Hsiang and Hsiang--Lawson on the construction of minimal submanifolds using equivariant geometry, and some extensions of their theory including new existence and compactness results. The major difference in our framework is that we do not require any "leftover" Killing vector fields in the conformal orbit spaces. As an application, we find 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.