Sessile droplet evaporation is widely encountered in nature, and it has numerous applications in industrial and scientific communities; therefore, the accurate prediction of droplet evaporation has great significance in practical applications. In this paper, for the first time, a comprehensive theoretical model is built up for diffusion-controlled heat and mass transfer for sessile droplet evaporation on a curved substrate in toroidal coordinates. The evaporative mass transfer is coupled with the heat transfer across the gas-liquid droplet interface, as well as the heat transfer across the solid-liquid interface of the curved substrate. The effects of interfacial cooling and thermal conductivity of the droplet and substrate as well as their initial shapes on the droplet evaporation are provided in details. It is found that the evaporative flux usually increases sharply near the droplet edge due to the short distance for heat conduction from the substrate to the droplet; however, it can be reversed from sharp increasing to decreasing at a low thermal conductivity ratio k(R) < 0.3 of the substrate over droplet or large initial droplet contact angle theta > 30 degrees. The interfacial evaporative cooling effect can always suppress the droplet evaporation. The lifetime of evaporative droplet can be prolonged with the decreasing thermal conductivity ratio, increasing evaporative cooling number, and increasing initial droplet contact angle or tangential angle of a curved substrate. These findings may be of great significance in the applications of droplet evaporation on the curved substrate.