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This chapter provides an overview of the formalism required to describe non-linear optical phenomena, including the non-linear wave equation and the different orders of susceptibilities, up to the third-order. Second- and third-order non-linear phenomena are reviewed, including second- and third-harmonic generation, as well as three-wave mixing. The remainder of the chapter is devoted to the description of important non-linear effects that can occur in nanostructures and stem either from surface or bulk non-linearities; the chapter concludes by discussing the enhancement mechanisms for the non-linear response of nanostructures.