  We study the probability distribution $F(u)$ of the maximum of smooth
Gaussian fields defined on compact subsets of $\R^d$ having some geometric
regularity.
  Our main result is a general formula for the density of $F$. Even though this
is an implicit formula, one can deduce from it explicit bounds for the density,
hence for the distribution, as well as improved expansions for $ 1-F(u)$ for
large values of $u$.
  The main tool is the Rice formula for the moments of the number of roots of a
random system of equation over the reals, of which we give a new simplified
proof.
  This method enables also to study second order properties of the so-called
expected Euler Characteristic approximation using only elementary arguments and
to extend these kind of results to some interesting classes of Gaussian fields.
  We obtain more precise results for the "direct method" to compute the
distribution of the maximum, using spectral theory of GOE random matrices.
