In this paper we study the conjugate heat transfer across a vertical wall separating two fluids at different temperatures. We present a classification of the solutions of the problem in terms of two main parameters; epsilon, the ratio of thickness to height of the wall and alpha, measuring the ratio of the thermal resistance of one of the boundary layers to the thermal resistance of the wail. Numerical and asymptotic solutions are presented for all possible values of alpha. We show that a maximum average Nusselt number or non-dimensional overall heat flux is attained for values of alpha much smaller than one, but still large compared with epsilon(2).