Content: Introduction Review of linear optics Induced polarization Harmonic oscillator model Local field corrections Estimated nonlinear response Summary Time-domain material response The polarization time-response function The Born-Oppenheimer approximation Raman scattering response function of silica Summary Material response in the frequency domain, susceptibility tensors The susceptibility tensor The induced polarization in the frequency domain Sum of monochromatic fields The prefactor to the induced polarization Third-order polarization in the Born-Oppenheimer approximation in the frequency domain Kramers-Kronig relations Summary Symmetries in nonlinear optics Spatial symmetries Second-order materials Third-order nonlinear materials Cyclic coordinate-system Contracted notation for second-order susceptibility tensors Summary The nonlinear wave equation Mono and quasi-monochromatic beams Plane waves - the transverse problem Waveguides Vectorial approach Nonlinear birefringence Summary Second-order nonlinear effects General theory Coupled wave theory Phase mismatch and acceptance bandwidths Second-harmonic generation Non-degenerate parametric frequency conversion Difference-frequency generation Frequency conversion of focused Gaussian beams Electro optic effects Summary Raman scattering Physical description Amplitude equations Fundamental characteristics of silica The Raman fiber amplifier Summary Brillouin Scattering Introduction Electrostriction Coupled wave equations Threshold Reduced SBS fibers Applications Summary Optical Kerr effect Short pulse propagation Propagation of short pulses Pulse characterization Applications of solitons and short pulse propagation Summary Four wave mixing Physical description Propagation equations - three frequencies Spontaneous emission in four-wave mixing Amplifiers Other Applications Summary A. Tensors B. Hamiltonian and polarization C. Signal analysis D. Generating matrices and susceptibility tensors E. Transverse field distributions F. The index ellipsoid G. Materials commonly used in nonlinear optics