  The aim of this paper is to develop foundations of umbral calculus on the
space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general
theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence
of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type,
and a Sheffer sequence. We present equivalent conditions for a sequence of
monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer
sequence, respectively. We also construct a lifting of a sequence of monic
polynomials on $\mathbb R$ of binomial type to a polynomial sequence of
binomial type on $\mathcal D'$, and a lifting of a Sheffer sequence on $\mathbb
R$ to a Sheffer sequence on $\mathcal D'$. Examples of lifted polynomial
sequences include the falling and rising factorials on $\mathcal D'$, Abel,
Hermite, Charlier, and Laguerre polynomials on $\mathcal D'$. Some of these
polynomials have already appeared in different branches of infinite dimensional
(stochastic) analysis and played there a fundamental role.
