The development of efficient methods to reject waste heat has been instrumental throughout aircraft design history. In current airliners heat exchangers cool the cabin air, avionics, and engine oil. As the aviation industry transitions toward electric or hydrogen-powered propulsion, heat exchangers will become more important. Electric aircraft must cool the batteries, motors, and other power electronics, which must be maintained at low operating temperatures. Hydrogen-powered aircraft need heat exchangers to maintain fuel cells' low operating temperature and to warm up liquid hydrogen fuel from cryogenic storage temperatures. The demands of next-generation aircraft drive the need for heat exchangers with low drag and weight. The current design methodology for heat exchangers remains rooted in analytical methods informed by empirical studies on simple geometries. This analysis method is limited to the range of geometries that can be analyzed and the availability of historical performance data. Numerical simulation tools can supplement analytical methods by providing a means to accurately assess the performance of arbitrary designs without the cost of an experimental study. Yet the cost to simulate the performance of each possible shape is prohibitively expensive. Numerical optimization alleviates this issue by using an optimization algorithm to investigate the design space. In particular, gradient-based optimization enables the inclusion of costly simulations due to a reduced number of iterations required to reach an optimum. Adjoint-based techniques, developed for aerodynamic shape optimization, compute the gradient information from fluid simulations used to predict the performance of heat exchangers. In this work, we applied adjoint-based shape optimization techniques to the design of heat exchangers. To support the adoption of shape optimization methods for heat exchanger design, we implemented our heat transfer analysis into the open-source optimization framework, MPhys. Furthermore, we investigated the combination of continuation techniques to form a novel method for globalizing a Newton solver to improve its robustness. To tune the developed method, we used a gradient-free optimization strategy to find the solver parameters that minimized the total wall time required to solve the test problems. The resulting method was faster and more robust than the individual continuation methods. Next, we demonstrated the merits of heat exchanger optimizations for two problems. First, we optimized the geometry of a plate-fin heat exchanger, which is prevalent in aviation, to minimize drag and mass. This work represents the first instance of adjoint-based shape optimization of a plate-fin heat exchanger and provides insight into the geometric characteristics needed to achieve each objective. The designs that minimized mass used wavy channels to produce separation and promote thermal mixing. Conversely, the designs that minimized drag used fewer long channels with a greater hydraulic diameter. Second, we applied the same methodology to the design of a surface heat exchanger of an electric aircraft motor. In this application, we compared the effect of the heat transfer model on the optimized design. Specifically, we compared a convective-only model to a conjugate model that couples convection and conduction. The results show that the optimizations with the conjugate model have less drag than those produced by the convection-only model due to its underprediction of heat transfer. In all, this work advances our ability to implement high-fidelity optimization approaches for the efficient design of heat exchangers. These approaches will become increasingly important as electric and hydrogen-powered aircraft rise in popularity.