We review the four main paradigms of tropical cyclone intensification that have emerged over the past five decades, discussing the relationship between them and highlighting their strengths and weaknesses. A major focus is on a new paradigm articulated in a series of recent papers using observations and highresolution, three-dimensional, numerical model simulations. Unlike the three previous paradigms, all of which assumed axial symmetry, the new one recognises the presence of localised, buoyant, rotating deep convection that grows in the rotation-rich environment of the incipient storm, thereby greatly amplifying the local vorticity. It exhibits also a degree of randomness that has implications for the predictability of local asymmetric features of the developing vortex. While surface moisture fluxes are required for intensification, the postulated 'evaporation-wind' feedback process that forms the basis of an earlier paradigm is not. Differences between spin up in three-dimensional and axisymmetric numerical models are discussed also.In all paradigms, the tangential winds above the boundary layer are amplified by the convectively-induced inflow in the lower troposphere in conjunction with the approximate material conservation of absolute angular momentum. This process acts also to broaden the outer circulation. Azimuthally-averaged fields from high-resolution model simulations have highlighted a second mechanism for amplifying the mean tangential winds. This mechanism, which is coupled to the first via boundary-layer dynamics, involves the convergence of absolute angular momentum within the boundary layer, where this quantity is not materially conserved, but where air parcels are displaced much further radially inwards than air parcels above the boundary layer. It explains why the maximum tangential winds occur in the boundary layer and accounts for the generation of supergradient wind speeds there. The boundary layer spin-up mechanism is not unique to tropical cyclones. It appears to be a feature of other rapidly-rotating atmospheric vortices such as tornadoes, waterspouts and dust devils and is manifest as a type of axisymmetric vortex breakdown. The mechanism for spin up above the boundary layer can be captured approximately by balance dynamics, while the boundary layer spin-up mechanism cannot. The spin-up process, as well as the structure of the mature vortex, are sensitive to the boundary-layer parameterisation used in the model.