We perform a detailed experimental study of large-scale vortices propagating in the rotating shallow-water layer in a paraboloidal vessel. A specific data acquisition technique is used in order to ensure precise measurements of the free-surface elevation. We find two qualitatively different types of vortex behaviour controlled by the relative elevation value. For small elevations we observe a standard quasi-geostrophic pattern with an asymmetric secondary circulation around an initially symmetric vortex which leads to a meridional drift and Rossby wave radiation. This type of behaviour is exhibited by both cyclonic and anticyclonic vortices. For relative elevations larger than 1 (nonlinear regime) the necessarily anticyclonic vortices are drifting strictly zonally maintaining their circular symmetry during the viscous decay. By varying the initial latitude of the vortex we were able to check that in the nonlinear regime the vortex lifetime is not sensitive to the beta-effect, while it is the case in the quasi-geostrophic regime. In the same way we show that the observed difference in cyclone–anticyclone lifetimes is not influenced by the beta-effect.