  Our goal in this paper is to extend the theory of quasi-exactly solvable
Schrodinger operators beyond the Lie-algebraic class. Let $\cP_n$ be the space
of n-th degree polynomials in one variable. We first analyze "exceptional
polynomial subspaces" which are those proper subspaces of $\cP_n$ invariant
under second order differential operators which do not preserve $\cP_n$. We
characterize the only possible exceptional subspaces of codimension one and we
describe the space of second order differential operators that leave these
subspaces invariant. We then use equivalence under changes of variable and
gauge transformations to achieve a complete classification of these new,
non-Lie algebraic Schrodinger operators. As an example, we discuss a finite gap
elliptic potential which does not belong to the Treibich-Verdier class.
