An analysis is made of the unsteady turbulent heat transfer in a circular tube whose wall temperature varies arbitrarily with lime. The flow is steady and fully developed. The formulation permits the heat-transfer coefficient to vary with time and position in accordance with the energy conservation principle. This is in contrast to previous transient analyses where it has been standard to use steady-state, fully developed coefficients. The first step in the analysis yields the heat-transfer response to a step jump in wall temperature, and this is then generalized by a superposition technique to apply to arbitrary thne variations. Use of the generalized results is illustrated by application to the case where the wall temperature varies linearly with time. Comparison is made between the unsteady heat-transfer results of the present theory and those computed using steady-state heat-transfer coefficients.