The discrete infinite-dimensional modal system describing nonlinear sloshing of an incompressible fluid with irrotational flow partially occupying a tank performing an arbitrary three-dimensional motion is derived in general form. The tank has vertical walls near the free surface and overturning waves are excluded. The derivation is based on the Bateman–Luke variational principle. The free surface motion and velocity potential are expanded in generalized Fourier series. The derived infinite-dimensional modal system couples generalized time-dependent coordinates of free surface elevation and the velocity potential. The procedure is not restricted by any order of smallness. The general multidimensional structure of the equations is approximated to analyse sloshing in a rectangular tank with finite water depth. The amplitude–frequency response is consistent with the fifth-order steady-state solutions by Waterhouse (1994). The theory is validated by new experimental results. It is shown that transients and associated nonlinear beating are important. An initial variation of excitation periods is more important than initial conditions. The theory is invalid when either the water depth is small or water impacts heavily on the tank ceiling. Alternative expressions for hydrodynamic loads are presented. The procedure facilitates simulations of a coupled vehicle–fluid system.