  This paper shows that there is a correspondence between quasi-exactly
solvable models in quantum mechanics and sets of orthogonal polynomials $\{
P_n\}$. The quantum-mechanical wave function is the generating function for the
$P_n (E)$, which are polynomials in the energy $E$. The condition of
quasi-exact solvability is reflected in the vanishing of the norm of all
polynomials whose index $n$ exceeds a critical value $J$. The zeros of the
critical polynomial $P_J(E)$ are the quasi-exact energy eigenvalues of the
system.
