In this chapter, we discuss the anharmonic phonon theory. The recent development of the first principles computational framework of phonon anharmonicity has enabled the calculation of lattice thermal conductivity, phonon lifetime, and other phonon-related properties. In the framework, harmonic and anharmonic interatomic force constants (IFCs) are extracted from first-principles density functional theory (DFT) or density functional perturbation theory (DFPT) calculations. Computing a dynamical matrix from harmonic IFCs yields frequencies and eigenvectors of ordinary harmonic phonons, whereas anharmonic IFCs determine self-energies that cause the frequency shifts and linewidths. We first introduce the classical lattice models, where we derive the complex normal coordinate expansion of the potential energy and dipole moment. Then, we quantize the complex normal coordinate and introduce the phonon operator. We further review the phonon Green’s function method, including the perturbation theory and the self-consistent phonon (SCPH) theory. Finally, we introduce the method to calculate the dielectric function.