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 mechanisms that occur in liquid crystals and photorefractive media. Additional nonlinear material systems and mechanisms such as glasses, molecular reorientation of single molecules, the electrostrictive effect, the nuclear effect (vibrational contributions), cascading, and the ever-present thermal effects are quantified, and representative tables of values are given.