Using Casorati determinants of Charlier polynomials, we construct for each finite set F of positive integers a sequence of polynomials rnF, n∈σF
Nearby in the stack
, which are eigenfunction of a second order difference operator, where
σF
is an infinite set of nonnegative integers,
σ_F
. For certain finite sets
F
(we call them admissible sets), we prove that the polynomials
rnF
,
n∈σF
, are actually exceptional Charlier 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 Charlier polynomials into a Wronskian determinant of Hermite polynomials. For admissible sets, these Wronskian determinants turn out to be exceptional Hermite polynomials.