Plasmonics has emerged as an important research field in nanoscience and nanotechnology. Recently, significant attention has been devoted to the observation and the understanding of nonlinear optical processes in plasmonic nanostructures, giving rise to the new research field called nonlinear plasmonics. This review provides a comprehensive insight into the physical mechanisms of one of these nonlinear optical processes, namely, second harmonic generation (SHG), with an emphasis on the main differences with the linear response of plasmonic nanostructures. The main applications, ranging from the nonlinear optical characterization of nanostructure shapes to the optimization of laser beams at the nanoscale, are summarized and discussed. Future directions and developments, made possible by the unique combination of SHG surface sensitivity and field enhancements associated with surface plasmon resonances, are also addressed.