  A general classification of linear differential and finite-difference
operators possessing a finite-dimensional invariant subspace with a polynomial
basis is given. The main result is that any operator with the above property
must have a representation as a polynomial element of the universal enveloping
algebra of the algebra of differential (difference) operators in
finite-dimensional representation. In one-dimensional case a classification is
given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$)
and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$),
$osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently,
$2 \times 2$ matrix operators in ${\bold R}$) and $gl_2 ({\bold R})_K$ ( for
the operators containing the differential operators and the parity operator). A
classification of linear operators possessing infinitely many
finite-dimensional invariant subspaces with a basis in polynomials is
presented.
