  A hypersurface $M$ in ${\bf R}^n$ or $S^n$ is said to be Dupin if along each
curvature surface, the corresponding principal curvature is constant. A Dupin
hypersurface is said to be proper Dupin if each principal curvature has
constant multiplicity on $M$, i.e., the number of distinct principal curvatures
is constant on $M$. The notions of Dupin and proper Dupin hypersurfaces in
${\bf R}^n$ or $S^n$ can be generalized to the setting of Lie sphere geometry,
and these properties are easily seen to be invariant under Lie sphere
transformations. This makes Lie sphere geometry an effective setting for the
study of Dupin hypersurfaces, and many classifications of proper Dupin
hypersurfaces have been obtained up to Lie sphere transformations. In these
notes, we give a detailed introduction to this method for studying Dupin
hypersurfaces in ${\bf R}^n$ or $S^n$, including proofs of several fundamental
results.
