  We emphasize intertwining relations as a universal tool in constructing
one-dimensional quasi-exactly solvable operators and offer their possible
generalization to the multidimensional case. Considered examples include all
quasi-exactly solvable operators with invariant subspaces of monomials. We show
that the simplest case of generalized intertwining relations allows to
naturally relate quasi-exactly solvable operators with two invariant monomial
subspaces to a nonlinear parasupersymmetry of second order. Quantum-mechanical
systems with linear and nonlinear supersymmetry are discussed from the
viewpoint of quasi-exact solvability. We construct such a general system with a
cubic supersymmetry and argue that quantum-mechanical systems with nonlinear
supersymmetry of fourth order and higher are generally not quasi-exactly
solvable. Besides, we construct two examples of quasi-exactly solvable
operators with invariant subspaces which cannot be reduced to monomial spaces.
