  In the past ten years, the ideas of supersymmetry have been profitably
applied to many nonrelativistic quantum mechanical problems. In particular,
there is now a much deeper understanding of why certain potentials are
analytically solvable and an array of powerful new approximation methods for
handling potentials which are not exactly solvable. In this report, we review
the theoretical formulation of supersymmetric quantum mechanics and discuss
many applications. Exactly solvable potentials can be understood in terms of a
few basic ideas which include supersymmetric partner potentials, shape
invariance and operator transformations. Familiar solvable potentials all have
the property of shape invariance. We describe new exactly solvable shape
invariant potentials which include the recently discovered self-similar
potentials as a special case. The connection between inverse scattering,
isospectral potentials and supersymmetric quantum mechanics is discussed and
multi-soliton solutions of the KdV equation are constructed. Approximation
methods are also discussed within the framework of supersymmetric quantum
mechanics and in particular it is shown that a supersymmetry inspired WKB
approximation is exact for a class of shape invariant potentials. Supersymmetry
ideas give particularly nice results for the tunneling rate in a double well
potential and for improving large $N$ expansions. We also discuss the problem
of a charged Dirac particle in an external magnetic field and other potentials
in terms of supersymmetric quantum mechanics. Finally, we discuss structures
more general than supersymmetric quantum mechanics such as parasupersymmetric
quantum mechanics in which there is a symmetry between a boson and a
para-fermion of order $p$.
