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We provide an in-depth treatment of the various mechanisms by which an incident light beam can produce an intensity- or flux-dependent change in the refractive index and absorption coefficient of different materials. Whenever possible, the mechanisms are initially traced to single-atom and -molecule effects in order to provide physical understanding. Representative values are given for the various mechanisms. Nine different mechanisms are discussed, starting with the Kerr effect due to atoms and/or molecules with discrete states, including organic materials such as molecules and conjugated polymers. Simplified two and/or three-level models provide useful information, and these are summarized. The nonlinear optics of semiconductors is reviewed for both bulk and quantum-confined semiconductors, focusing on the most common types II-VI and III-V. Also discussed in some detail are the different nonlinear …