Rapidly rotating convection in spherical geometry outside the tangent cylinder is investigated using the quasi-geostrophic approximation. The validity of the approximation is discussed, and numerical simulations using these equations are performed, reaching Ekman numbers, $E$, down to $10^{-6}$. The results are compared with experiments and fully three-dimensional numerical simulations. We find that the inertial scaling developed to study rapidly rotating convection does not represent the Prandtl-number dependence of our results adequately. Instead, we find that even in strongly supercritical situations the dominant wavenumbers at the onset of convection still have a strong influence on the behaviour. We find that the local Péclet number, the product of the typical convective velocity and local convective length scale divided by the thermal diffusivity, is helpful for understanding the dynamics of rapidly rotating convection. We explore the zonal flows driven by Reynolds stresses with no-slip boundaries and explore their Prandtl-number dependence. We also study the convective heat transport at low $E$, and consider the boundary layer structures that can form at large Rayleigh number, slowing down the rate of growth of the Nusselt number with Rayleigh number.