The principal goal of this thesis is to develop a systematic method for the design of composite heat sinks (CHSs) that serve as passive and transient cooling devices for microelectronics. This is accomplished by posing the CHS design problem as a topology optimization problem wherein a phase-change material and a high-conductivity material are to be optimally distributed. Two different types of formulations are proposed. The first one aims to maximize the time of operation before a tolerable temperature is reached at the interface between a heat source and the CHS. The second one aims to minimize the maximum temperature across the heating interface for a given time of operation. The two materials are interpolated in topology optimization using the usual mixture law with penalty. The phase-change is modeled using the apparent heat capacity method in which the specific heat is taken as a nonlinear function of the temperature so that the latent heat absorption is accounted for at the melting point. The ensuing new transient topology optimization problem involving an interpolated material property that depends on the state variable is solved using continuous optimization algorithm. The validity of the phase-change modeling is verified with a one dimensional model as well as experimentation. Analytical sensitivity analysis is derived and verified with the finite difference derivatives. Several examples are solved to illustrate the intricacies of the problem and the effectiveness and the limitations of the proposed design method. Prototypes of an intuitively conceived CHS and optimized one are made. An experimental setup is devised to test the two prototypes. Based on the insight gained from the experiments, an improved conduction model is studied to also incorporate convective heat transfer also into the model.