  In recent years, one of the most interesting developments in quantum
mechanics has been the construction of new exactly solvable potentials
connected with the appearance of families of exceptional orthogonal polynomials
(EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre
and Hermite) classical orthogonal polynomials, which start with a constant, the
EOP families begin with some polynomial of degree greater than or equal to one,
but still form complete, orthogonal sets with respect to some positive-definite
measure. We show how they may appear in the bound-state wavefunctions of some
rational extensions of well-known exactly solvable quantum potentials. Such
rational extensions are most easily constructed in the framework of
supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class
of translationally shape invariant potentials. We review the most recent
results in this field, which use higher-order SUSYQM. We also comment on some
recent re-examinations of the shape invariance condition, which are independent
of the EOP construction problem.
