Nonlinearity causes equatorially-trapped Rossby waves to form anticyclonic, contra-rotating vortex pairs. We describe recent studies on vortex stability, vorticity and streamfunction relationships, and higher mode weakly nonlocal solitons. The Kelvin wave may similarly form solitary waves in the presence of dispersion-inducing mean currents. At sufficiently large amplitude, however, the Kelvin wave will break. We describe recent work on the limiting soliton, its connection with so-called "corner waves" in other wave systems, and also on the weakly nonlocal fronts that form if breaking is thwarted by viscosity.