Topology optimization is a versatile design tool for the synthesis of heterogeneous engineering systems where the optimal distribution of constituent materials is sought such that a prescribed measure of performance is optimized. In this dissertation, topology optimization methodologies are developed for solving problems associated with wave propagation and vibration in elastic and piezoelectric media. These methodologies utilize the finite element method in conjunction with gradient-based optimization algorithms to create functional materials, structures, and devices. The methodologies are demonstrated in a number of examples and illustrative studies that progress the state-of-the-art in the fields of topology optimization, elastic waveguides, phononic band-gap materials, and piezoelectric energy harvesting systems. These include the design of bulk and surface wave elastic waveguides in two and three dimensions that guide various forms of wave energy as desired, band-gap structures that provide tailored frequency transmission spectrums for bulk waves and surface waves, band-gap materials that prevent wave propagation within certain frequencies, and piezoelectric energy harvesting systems designed to optimize power output. Also addressed are previously unreported issues with the application of topology optimization to these types of problems including the role of physical phenomena in the solutions, mesh dependency effects, non-uniqueness, and the impact of small feature sizes.