Previous investigations have been concerned with the stress concentration around a spheroidal cavity under certain restricted loading assumptions. In this paper exact closed solutions are obtained for the following: (a) The distribution of stress around a spheroidal inclusion in an elastic body which is in an arbitrary uniform state of stress infinitely far from the inclusion, and, in particular, for the distribution of stress around a spheroidal cavity under the same loading conditions. (b) The thermal stress distribution arising from distinct uniform temperature changes applied to the spheroidal inclusion and the surrounding medium. Technically important features of the solutions obtained are discussed.