Wherever friction occurs, mechanical energy is transformed into heat. The maximum surface temperature associated with this heating can have an important influence on the tribological behavior of the contacting components. For band contacts the partitioning of heat has already been studied extensively; however, for circular and elliptic contacts only approximate solutions exist. In this work a numerical algorithm is described to solve the steady state heat partitioning and the associated flash temperatures for arbitrary shaped contacts by matching the surface temperatures of the two contacting solids at all points inside the contact area. For uniform and semi-ellipsoidal shaped heat source distributions, representing EHL conditions and dry or boundary lubrication conditions respectively, function fits for practical use are presented giving the flash temperature as a function of the Pe´clet numbers of the contacting solids, the conductivity ratio, and the aspect ratio of the contact ellipse. These function fits are based on asymptotic solutions for small and large Pe´clet numbers and are valid for the entire range of Pe´clet numbers. By comparison with numerical results they are shown to be accurate within 5%, even for the situation of opposing surface velocities.