Summary: Phononic band-gap materials prevent elastic waves in certain frequency ranges from propagating, and they may therefore be used to generate frequency filters, as beam splitters, as sound or vibration protection devices, or as waveguides. In this work we show how topology optimization can be used to design and optimize periodic materials and structures exhibiting phononic band gaps. Firstly, we optimize infinitely periodic band-gap materials by maximizing the relative size of band gaps. Then, finite structures subjected to periodic loading are optimized in order to either minimize the structural response along boundaries (wave damping) or maximize the response at certain boundary locations (waveguiding).