A method is presented for calculating the development of momentum and thermal laminar boundary layers on a heated cylinder of arbitrary shape when the cylinder moves in an incompressible fluid at rest with an unsteady velocity. This analysis is based on solutions to the unsteady momentum and energy-integral equations in conjunction with a set of universal functions, derived from exact solutions to the boundary-layer equations for a specific unsteady problem. These universal functions are given in tabulated form. Those associated with the energy-integral equation are calculated with a Prandtl number of 0.7. The reliability and limitation of these functions are indicated and discussed in the light of several simple problems of which solutions are available. A detailed calculation procedure for the general unsteady problem is given and then followed by a numerical example.