Rapidly rotating Rayleigh–Bénard convection is studied using an asymptotically reduced equation set valid in the limit of low Rossby numbers. Four distinct dynamical regimes are identified: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Rayleigh numbers, followed for yet larger Rayleigh numbers by a breakdown of the convective Taylor columns into a disordered plume regime characterized by reduced efficiency and finally by geostrophic turbulence. The transitions are quantified by examining the properties of the horizontally and temporally averaged temperature and thermal dissipation rate. The maximum of the thermal dissipation rate is used to define the width of the thermal boundary layer. In contrast to the non-rotating Rayleigh–Bénard convection, the temperature drop across this layer decreases monotonically with increasing Rayleigh number and does not saturate. The breakdown of the convective Taylor column regime is attributed to the onset of convective instability of the thermal boundary layer and confirmed using the explicit linear stability analysis. Horizontal spectra of the vorticity, vertical velocity and temperature fluctuations are computed and their evolution with time is elucidated. A large-scale barotropic mode evolves from random initial conditions on an extremely long time scale and leads to continued evolution of the nominally saturated Nusselt number and its variance over very long times. The results are used to provide insights into the dynamics of rapidly rotating convection outside the asymptotic regime described by the reduced equations.