The topology optimization of systems subject to heat and mass transfers shows a wide potential for designing optimal and innovative structures in energy engineering. The present works apply the concepts of the homogenization method to two configurations subject to conductive and convective heat transfers. After recalling some basic principles, a special attention is given to the numerical approach dealing with multi-objective optimization problems that naturally occurs in several practical case studies. The two last sections are dedicated to some numerical investigations and provide a physical interpretation of the structural features reached by the optimization process. The main conclusion deals with the possibility of finding an acceptable trade-off between different objective functions, for both conductive and convective shape optimization problems, provided that the Pareto frontier is convex.