  Using Casorati determinants of Meixner polynomials $(m_n^{a,c})_n$, we
construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers a
sequence of polynomials $m_n^{a,c;\F}$, $n\in \sigma_\F$, which are
eigenfunctions of a second order difference operator, where $\sigma_\F$ is
certain infinite set of nonnegative integers, $\sigma_\F \varsubsetneq \NN$.
When $c$ and $\F$ satisfy a suitable admissibility condition, we prove that the
polynomials $m_n^{a,c;\F}$, $n\in \sigma_\F$, are actually exceptional Meixner
polynomials; that is, in addition, they are orthogonal and complete with
respect to a positive measure. By passing to the limit, we transform the
Casorati determinant of Meixner polynomials into a Wronskian type determinant
of Laguerre polynomials $(L_n^\alpha)_n$. Under the admissibility conditions
for $\F$ and $\alpha$, these Wronskian type determinants turn out to be
exceptional Laguerre polynomials.
