We consider an orthogonal system of stochastic polynomials with respect to a Lévy stochastic measure on a general topological space. In the case the stochastic measure is Gaussian or of the Poisson type, this orthogonal system turns out to have properties similar to the ones of the Hermite polynomials of Gaussian variables. We also deal with stochastic differentiation with respect to Lévy stochastic measures on topological spaces. We introduce a version of the Malliavin derivative and we suggest a direct differentiation formula which is valid for all stochastic polynomials.