Heat exchangers have long been used in a wide variety of industrial applications, such as for energy recovery from by-products, temperature regulation in chemical processes, refrigeration, or cooling of car engines. Typically, each application requires a different type of heat exchanger such as, tube, shell, with/without phase change, mixing/non mixing etc. heat exchangers. Due to their importance, there has been an ongoing interest in reducing the perational/constructional costs and increasing the efficiency. A lot of research has be done in optimizing certain features of heat exchangers (e.g. tube dimensions, fin thickness etc.), but so far none of them investigates the optimization of the whole topology of a heat exchanger. The aim of this thesis is to optimize the structure of a two flow heat exchanger, by means of topology optimization. More specifically we aim to maximize the efficiency of heat transfer, given some predefined pressure drop and dimension constraints. These constraints are necessitated by the need of achieving a reduced operating (pressure drop) and manufacturing dimensions) costs. A heat exchanger, being a multi-physics system, can be described by two physical phenomena: the flow of the fluid and the heat transfer. In this study we focus on heat exchanger governed by an isothermal and incompressible Stokes flow with low Reynolds number, while the heat transfer is assumed to be advective-conductive heat transfer, without internal heat generation, characterised by a relatively high Peclet number. We evaluate two novel models for topology optimization of heat exchangers; the Fluid Tracking Model and the Multi-Material Model. Throughout the experimental evaluation we saw that the Multi-Material Model performs best. The Fluid Tracking Model did not produce optimal results and was unable to enforce non-mixing designs. The Multi-Material Model optimized designs that maximized the heat transfer surface area between the fluids. Furthermore the designs illustrated a wall at the interfaces of the two fluids, keeping the two flows separated. Both 2D and 3D cases were studied. The 3D optimal results achieved a moderate improvement in performance over a simple design of a concentric tube heat exchanger.