Isoparametric hypersurfaces in the sphere are those whose
principal curvatures are everywhere constant with fixed multiplicities. In
some sense, such hypersurfaces represent the simplest type of manifolds we
can get a handle on. They have rather complicated topology and most of them
are inhomogeneous, and thus they serve as a good testing ground for
constructing examples and counterexamples. The classification of such
hypersurfaces was initiated by E. Cartan around 1938, and the completion of
the last case with four principal curvatures will appear soon in
publication. Since the classification is a long story covering a wide
spectrum of mathematics, I will highlight in this talk the decisive moments
and the key ideas engaged in the intriguing pursuit.