The realization of strong interactions between individual photons is a long-standing goal of both fundamental and technological significance. Scientists have known for over half a century that light fields can interact inside nonlinear optical media, but the nonlinearity of conventional materials is negligible at the light powers associated with individual photons. Nevertheless, remarkable advances in quantum optics have recently culminated in the demonstration of several methods for generating optical nonlinearities at the level of individual photons. Systems exhibiting strong photon-photon interactions enable a number of unique applications, including quantum-by-quantum control of light fields, single-photon switches and transistors, all-optical deterministic quantum logic, and the realization of strongly correlated states of light and matter.