Conventional boundary-layer theory cannot be applied when the fluid velocity outside the layer changes direction, and the leading edge of a finite body changes ends. In this paper an approximate method for examining the details of the boundary layer during a single flow reversal (occurring at t = 0) is described. It is based on the expectation that (a) long before reversal (t < -t1), there will be a quasi-steady boundary layer appropriate to flow in one direction; (b) long after reversal (t > t2) there will be a quasi-steady boundary layer appropriate to flow in the opposite direction, and (c) in between there will be a period of pure diffusion. The method is applied to a simple heat-transfer problem, in which a fluid of thermal diffusivity D flows with uniform velocity U = At over the plane y = 0; the strip 0 < x < L of the plane is maintained at temperature T1, while the restof the plane and the fluid far away have temperature T0. The approximate solution is compared with an exact solution of the boundary-layer equation, and is shown to give an accurate prediction of the heat transfer as a function of time. The boundary-layer approximation itself breaks down in regions of length O(D2/3A−1/3) near the ends of the heated strip, as usual; it also breaks down in the neighbourhood of the point x = 1/2At2, t > 0, at which the influence of the new leading edge is first felt after flow reversal. In a solution of the full equation, this region is examined in detail, and boundary-layer theory is shown to be sufficiently accurate for the calculation of heat transfer.