The unsteady, laminar, free-convection boundary layer is analyzed with the Grashof number considered to be time-dependent through either the uniform wall temperature or the acceleration field. Two geometries are considered; the vertical plate and the horizontal circular cylinder. A set of parameters is derived through which it is possible to describe the unsteady behavior of the boundary layer. These parameters allow solutions of the pertinent differential equations to be expressed in series form and an infinite number of sets of perturbation equations result. Numerical integrations of the first few sets are shown graphically. These results enable one to visualize and to calculate, quite readily, the time-dependent deviations from the quasi-steady velocity and temperature profiles as well as the deviation in heat transfer.