Near resonance sloshing in containers, filled with a liquid to a given depth h, depends on three parameters, which are the viscous damping, the frequency offset that contains the forcing amplitude and the fluid depth. Experiments have been conducted with low-viscosity liquids mainly in circular cylindrical containers of radius R subjected to harmonic horizontal forcing; complementary experiments on wave breaking have been performed in a square-base container. The fluid depth was kept large (h/R > 1) so that it was no longer a variable parameter. The bounds of existence of the different wave regimes, namely planar waves, swirling waves, chaotic sloshing as well as breaking waves, have been determined as a function of forcing frequencies relative to the lowest natural frequency ω1 and for a wide range of forcing amplitudes. It is shown that when the forcing frequency ω is slightly larger than the lowest natural frequency ω1, planar wave motion bifurcates to a swirling wave mode at finite wave amplitude, the value of which depends on the offset parameter. The swirl wave amplitude grows exponentially and saturates at a certain value. The swirl has a hard-spring behaviour, is very robust and can generate a vortical flow of the liquid column. Chaotic sloshing and wave breaking occur quasi-periodically: growth of planar wave amplitude at a rate depending on the forcing amplitude, collapse, irregular swirl and again growth of planar wave amplitude. The details and periodicity of the chaotic behaviour and breaking depend on the frequency-offset parameter. Close to the natural frequency, chaotic wave motion is possible without breaking. Planar wave breaking is, in general, associated with spilling caused by the encounter of nearly freely falling lumps of fluid with the upward moving wave crest, in a way demonstrated previously in two-dimensional wave breaking. In three dimensions, the wave crest is destabilized in the crosswise direction so that spilling is not uniform along the wave crest and an irregular swirl is generated following breaking; free fall of fluid lumps occurs over many wave periods. The complementary experiments, performed in a square-base container of base dimension L, show four different wave patterns of wavelengths L and L/2 crosswise to the primary wave. This cross-wave instability is interpreted in terms of parametric instability.