  Let $f$ be a Gaussian random field on $\mathbb{R}^d$ and let $X$ be the
number of critical points of $f$ contained in a compact subset. A long-standing
conjecture is that, under mild regularity and non-degeneracy conditions on $f$,
the random variable $X$ has finite moments. So far, this has been established
only for moments of order lower than three. In this paper, we prove the
conjecture. Precisely, we show that $X$ has finite moment of order $p$, as soon
as, at any given point, the Taylor polynomial of order $p$ of $f$ is
non-degenerate. We present a simple and general approach that is not specific
to critical points and we provide various applications. In particular, we show
the finiteness of moments of the nodal volumes and the number of critical
points of a large class of smooth, or holomorphic, Gaussian fields, including
the Bargmann-Fock ensemble.
