An analysis is made for laminar forced-convection heat transfer in a circular tube or a parallel plate channel whose walls may undergo arbitrary time variations in temperature. The time-varying process can begin from an already established steady-state situation with heat transfer taking place, or the fluid and walls can be initially at the same uniform temperature. The fluid velocity distribution is fully developed and unchanging with time. At any instant during the transient the wall temperature is spatially uniform, that is, all portions of the wall simultaneously undergo the same temperature-time variation. The greater part of the analysis is concerned with the response to a step change in wall temperature, and the time required to reach steady state is given for this type of transient. Then the results are generalized to apply for arbitrary variations with time.