  A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is
isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and
$H^n$, the classification of isoparametric hypersurfaces is complete and
relatively simple, but as Elie Cartan showed in a series of four papers in
1938-1940, the subject is much deeper and more complex for hypersurfaces in the
sphere $S^n$. A hypersurface $M^{n-1}$ in a real space-form is proper Dupin if
the number $g$ of distinct principal curvatures is constant on $M^{n-1}$, and
each principal curvature function is constant along each leaf of its
corresponding principal foliation. This is an important generalization of the
isoparametric property that has its roots in nineteenth century differential
geometry and has been studied effectively in the context of Lie sphere
geometry. This paper is a survey of the known results in these fields with
emphasis on results that have been obtained in more recent years and discussion
of important open problems in the field.
