Classifications of Dupin Hypersurfaces in Lie Sphere Geometry

This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\mathbf{R}}^n$) 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 ${\mathbf{R}}^n$), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.