A hypersurface $M^n$ in a real space form ${ \bf R}^{ n+1}$, $S^{ n+1}$, or $H^{ n+1}$ is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and M\"{ u}nzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938--1940, and two papers of M\"{ u}nzner that were published in preprint form in the early 1970's, and as journal articles in 1980--1981. These papers of Cartan and M\"{ u}nzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres.
Categories: math.DG