  We combine Malliavin calculus with Stein's method to derive bounds for the
Variance-Gamma approximation of functionals of isonormal Gaussian processes, in
particular of random variables living inside a fixed Wiener chaos induced by
such a process. The bounds are presented in terms of Malliavin operators and
norms of contractions. We show that a sequence of distributions of random
variables in the second Wiener chaos converges to a Variance-Gamma distribution
if and only if their moments of order two to six converge to that of a
Variance-Gamma distributed random variable (six moment theorem). Moreover,
simplified versions for Laplace or symmetrized Gamma distributions are
presented. Also multivariate extensions and a universality result for
homogeneous sums are considered.
