  We compute the probability distribution of the interface width at the
depinning threshold, using recent powerful algorithms. It confirms the
universality classes found previously. In all cases, the distribution is
surprisingly well approximated by a generalized Gaussian theory of independant
modes which decay with a characteristic propagator G(q)=1/q^(d+2 zeta); zeta,
the roughness exponent, is computed independently. A functional renormalization
analysis explains this result and allows to compute the small deviations, i.e.
a universal kurtosis ratio, in agreement with numerics. We stress the
importance of the Gaussian theory to interpret numerical data and experiments.
