  This paper summarizes some known results about Appell polynomials and
investigates their various analogs. The primary of these are the free Appell
polynomials. In the multivariate case, they can be considered as natural
analogs of the Appell polynomials among polynomials in non-commuting variables.
They also fit well into the framework of free probability. For the free Appell
polynomials, a number of combinatorial and "diagram" formulas are proven, such
as the formulas for their linearization coefficients. An explicit formula for
their generating function is obtained. These polynomials are also martingales
for free Levy processes. For more general free Sheffer families, a necessary
condition for pseudo-orthogonality is given. Another family investigated are
the Kailath-Segall polynomials. These are multivariate polynomials, which share
with the Appell polynomials nice combinatorial properties, but are always
orthogonal. Their origins lie in the Fock space representations, or in the
theory of multiple stochastic integrals. Diagram formulas are proven for these
polynomials as well, even in the q-deformed case.
