Large, long-lived vortices, surviving during many turnaround times and far longer than the dispersive linear Rossby wave packets, are abundant in planetary atmospheres and oceans. Nonlinear effects which prevent dispersive decay of intense cyclones and anticyclones and provide their self-propelling propagation are revised here using shallow water equations and their balanced approximations. The main physical mechanism allowing vortical structures to be long-lived in planetary fluid is the quick fluid rotation inside their cores which prevents growth in the amplitude of asymmetric circulation arising due to the beta-effect. Intense vortices of both signs survive essentially longer than the linear Rossby wave packet if their azimuthal velocity is much larger than the Rossby wave speed. However, in the long-time evolution, cyclonic and anticyclonic vortices behave essentially differently that is illustrated by the conservative intermediate geostrophic model. Asymmetric circulation governing vortex propagation is described by the azimuthal mode m=1 for the initial value problem as well as for steadily propagating solutions. Cyclonic vortices move west-poleward decaying gradually due to Rossby wave radiation while anticyclonic ones adjust to non-radiating solitary vortices. Slow weakening of an intense cyclone with decreasing of its size and shrinking of the core is described assuming zero azimuthal velocity outside the core while drifting poleward. The poleward tendency of the cyclone motion relative to the stirring flow corresponds to characteristic trajectories of tropical cyclones in the Earth's atmosphere. The asymmetry in dispersion-nonlinear properties of cyclones and anticyclones is thought to be one of the essential reasons for the observed predominance of anticyclones among long-lived vortices in the atmospheres of the giant planets and also among intrathermoclinic eddies in the ocean.