Phononic crystals are periodic composite structures where the elastic and acoustic properties display a periodic variation in space. They are in general constituted by a periodic array of inclusions in a matrix. Due to their periodicity, they can exhibit absolute band gaps in which the propagation of elastic waves is prohibited in any direction of the space. More generally, tailoring their band structure allows the control and manipulation of elastic waves and pave the way to several functionalities ranging from sound isolation to filtering and signal processing, negative refraction and high resolution imaging, nanoscale thermal transport managing, quantum information processing. In this chapter, we discuss some basic properties of the phononic crystals, in particular the dependence of the band gaps with the nature of the constituent materials (solid or fluid), the contrast between the elastic properties of the constituents, the shape and the filling fraction of the inclusions, the crystal lattice. We discuss the existence of gaps resulting from Bragg interference or from local resonances. The localized modes associated with defects such as cavities and waveguides are presented and their functionalities in filtering and multiplexing applications are discussed.