An analysis is made of the unsteady thermal processes which result when a moving sphere encounters a fluid environment whose initial temperature is different from that of the sphere. The fluid velocity field is steady and Stokesian, and the temperature field is of the boundary-layer type. Two related physical situations are analyzed. In one of these, the sphere is a solid of high thermal conductance such that its temperature is spatially uniform, but varies with time as a result of heat transfer with the fluid. In the other, the sphere heat capacity and conductance are both large so that its surface temperature is spatially and temporally uniform for a finite period of time. A number of solution methods is employed, including numerical inversion of integral transforms, series, and asymptotic expansions. Results are presented both in graphical and algebraic form for the local and overall heat-transfer rates, for the temperature history, and and for the time required to reach thermal equilibrium.