In this paper, we review the theory of time space-harmonic polynomials
developed by using a symbolic device known in the literature as the classical
umbral calculus. The advantage of this symbolic tool is twofold. First a moment
representation is allowed for a wide class of polynomial stochastic involving
the L\'evy processes in respect to which they are martingales. This
representation includes some well-known examples such as Hermite polynomials in
connection with Brownian motion. As a consequence, characterizations of many
other families of polynomials having the time space-harmonic property can be
recovered via the symbolic moment representation. New relations with
Kailath-Segall polynomials are stated. Secondly the generalization to the
multivariable framework is straightforward. Connections with cumulants and Bell
polynomials are highlighted both in the univariate case and in the multivariate
one. Open problems are addressed at the end of the paper
