We introduce a class of orthogonal polynomials in two variables which
generalizes the disc polynomials and the 2-D Hermite polynomials. We identify
certain interesting members of this class including a one variable
generalization of the 2-D Hermite polynomials and a two variable extension of
the Zernike or disc polynomials. We also give q-analogues of all these
extensions. In each case in addition to generating functions and three term
recursions we provide raising and lowering operators and show that the
polynomials are eigenfunctions of second-order partial differential or
q-difference operators