Coriolis-centrifugal convection (C³), i.e. rotating thermal convection including centrifugal buoyancy, constitutes an idealised, yet self-consistent, model for tornadoes, hurricanes, and typhoons. In this talk, I will review the possible C³ regimes and their relevance for natural storm systems based on a suite of direct numerical simulations in cylindrical domains. In the regime where the flow is in a triple balance between pressure gradient, Coriolis, and centrifugal force, hurricane- and typhoon-like structures form, such as eyes and secondary eyewalls. In the regime where the dominant dynamical balance is quasicyclostrophic, that is between the centrifugal buoyancy and the pressure gradient forces, tornado-like vortices are generated. In this case, the rotating cylinder in the simulations represents the mesocyclone of a supercell thunderstorm. I will discuss the tornado features that are captured, including the azimuthal velocity profiles, intensification of the vortex strength, and helicity characteristics. Furthermore, I will present a possible explanations for why seemingly similar mesocyclones may or may not spawn tornadoes, which is consistent with recent observational estimates.