Brillouin spectroscopy and imaging are emerging techniques in analytical science, biophotonics, and biomedicine. They are based on Brillouin light scattering from acoustic waves or phonons in the GHz range, providing a nondestructive contactless probe of the mechanics on a microscale. Novel approaches and applications of these techniques to the field of biomedical sciences are discussed, highlighting the theoretical foundations and experimental methods that have been developed to date. Acknowledging that this is a fast moving field, a comprehensive account of the relevant literature is critically assessed here.