Lattice vibrations are collective excitations of a crystal. This chapter considers elastic vibrations in a crystal with one atom per cell. It discusses a system of two atoms per cell. For each polarization mode in a given direction, the dispersion relation ω(K) develops two types of branches. These branches are acoustic and optical. Each branch can have longitudinal as well as transverse modes. The chapter also discusses the flow of thermal energy due to phonons. Neutrons interact with a crystal by the scattering from the atomic nuclei and thus carry information about the vibrational properties of the crystal. In order to obtain the number of modes at a given frequency range, the chapter considers phonon modes to be similar to vibrational modes on a one-dimensional string at a particular frequency. The Debye temperature is a measure of the maximum vibrational energy of a crystal.