We present in this paper a rather general method for the construction of so
-called conditionally exactly solvable potentials. This method is based on
algebraic tools known from supersymmetric quantum mechanics. Various famili
es of one-dimensional potentials are constructed whose corresponding Schrod
inger eigenvalue problem can be solved exactly under certain conditions of
the potential parameters. Examples of quantum systems on the real line and
the half line as well as on some finite interval are studied in detail. (C)
1998 Academic Press.