  Starting from the moment sequences of classical orthogonal polynomials we
derive the orthogonality purely algebraically. We consider also the moments of
($q=1$) classical orthogonal polynomials, and study those cases in which the
exponential generating function has a nice form. In the opposite direction, we
show that the generalized Dumont-Foata polynomials with six parameters are the
moments of rescaled continuous dual Hahn polynomials. Finally we show that one
of our methods can be applied to deal with the moments of Askey-Wilson
polynomials.
