This paper documents the conjugate heat transfer through a wall with nonuniform thickness, which is lined on one side by a boundary layer. In the first part, variational calculus shows that the total heat transfer rate is minimized w hen the wall thickness decreases in an optimal manner in the direction of flow. The reductions in total he; t transfer rate are significant when the Biot number is smaller than 1. In the second part of the study, th e complete problem of a laminar forced convection boundary layer coupled with conduction through a variiible-thickness wall is solved numerically. Means for calculating the total heat transfer rate are reported graphically. It was again found that the total heat transfer rate decreases when the wall profile is tapered so that the wall thickness decreases in the direction of flow. © 1992.