  We consider orthogonal polynomials with respect to a linear differential
operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$
where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq
k, 0\leq k \leq M$, with equality for at least one index. We analyze the
uniqueness and zero location of these polynomials. An interesting phenomenon
occurring in this kind of orthogonality is the existence of operators for which
the associated sequence of orthogonal polynomials reduces to a finite set. For
a given operator, we find a classification of the measures for which it is
possible to guarantee the existence of an infinite sequence of orthogonal
polynomials, in terms of a linear system of difference equations with varying
coefficients. Also, for the case of a first-order differential operator, we
locate the zeros and establish the strong asymptotic behavior of these
polynomials.
