  In this paper I prove good estimates on the moments and tail distribution of
$k$-fold Wiener--It\^o integrals and also present their natural counterpart for
polynomials of independent Gaussian random variables. The proof is based on the
so-called diagram formula for Wiener--It\^o integrals which yields a good
representation for their products as a sum of such integrals. I intend to show
in a subsequent paper that this method also yields good estimates for
degenerate $U$-statistics. The main result of this paper is a generalization of
the estimates of Hanson and Wright about bilinear forms of independent standard
normal random variables. On the other hand, it is a weaker estimate than the
main result of a paper of Lata{\l}a [6]. But that paper contains an error, and
it is not clear whether its result is true. This question is also discussed
here.
