  This is a survey of local and global classification results concerning Dupin
hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context
of Lie sphere geometry. The emphasis is on results that relate Dupin
hypersurfaces to isoparametric hypersurfaces in spheres. Along with these
classification results, many important concepts from Lie sphere geometry, such
as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of
$S^n$ (or ${\bf R}^n$), are described in detail. The paper also contains
several important constructions of Dupin hypersurfaces with certain special
properties.
