In this paper we formulate three inverse design, isoperimetric, shape optimization problems leading to optimum geometries that maximize the heat transfer rate. We examine both semi-infinite and finite, two-dimensional domains. The former is bounded from below by an isothermal periodic boundary while a constant heat flux is assumed at the far field; the latter is bounded by a periodic boundary at the bottom and a flat isothermal surface at the top. The objective is to find the optimum shape of the periodic boundary such that the heat transfer rate is maximized. We consider three different applications: (i) the optimal shape of corrugations (surface "roughness"), (ii) the optimal shape of high conductivity inserts (inverted fins) and (iii) the optimal shape of high conductivity fins. As expected, the optimum geometries have the shape of a depression, however the details of the geometries are quite interesting. In particular, the shape optimization problem associated with the inverse design of a high-conductivity extended surface/fin has led to some interesting results, with practical implications. (c) 2013 Elsevier Ltd. All rights reserved.