Summary: We study here the asymptotic behaviour of weakly transverse water-waves in the long waves regime. It is well-known that the Kadomtsev-Petviashvili (KP) approximation describes formally the dynamics of the exact solutions of the water-waves equations. We provide here a rigorous justification of this approximation, showing that if solutions of the water-waves equations exist over a relevant time scale, then they are well approximated by the KP approximation. A nonphysical zero mass assumption, inherent to the structure of the KP equation, is however needed to obtain this result; this is the reason why we introduce a class of weakly transverse Boussinesq systems. These new systems provide a much more precise approximation than the KP equation and do not require any zero mass assumption.