A thermodynamic approach to estimating maximum potential intensity (MPI) of tropical cyclones is described and compared with observations and previous studies. The approach requires an atmospheric temperature sounding, SST, and surface pressure; includes the oceanic feedback of increasing moist entropy associated with falling surface pressure over a steady SST; and explicitly incorporates a cloudy eyewall and a clear eye. Energetically consistent, analytic solutions exist for all known atmospheric conditions. The method is straightforward to apply and is applicable to operational analyses and numerical model forecasts, including climate model simulations. The derived MPI is highly sensitive to the surface relative humidity under the eyewall, to the height of the warm core, and to transient changes of ocean surface temperature. The role of the ocean is to initially contribute to the establishment of the ambient environment suitable for cyclone development, then to provide the additional energy required for development of an intense cyclone. The major limiting factor on cyclone intensity is the height and amplitude of the warm core that can develop; this is closely linked to the height to which eyewall clouds can reach, which is related to the level of moist entropy that can be achieved from ocean interactions under the eyewall. Moist ascent provides almost all the warming above 200 hPa throughout the cyclone core, including the eye, where warm temperatures are derived by inward advection and detrainment mixing from the eyewall. The clear eye contributes roughly half the total warming below 300 hPa and produces a less intense cyclone than could be achieved by purely saturated moist processes. There are necessarily several simplifications incorporated to arrive at a tractable solution, the consequences of which are discussed in detail. Nevertheless, application of the method indicates very close agreement with observations. For SST < 26°C there is generally insufficient energy for development. From 26° to 28°C SST the ambient atmosphere warms sharply in the lower troposphere and cools near the tropopause, but with little change in midlevels. The result is a rapid increase of MPI of about 30 hPa °C−1. At higher SST, the atmospheric destabilization ceases and the rate of increase of MPI is reduced.