In this chapter we describe the basic concepts of two-dimensional (2D) phononic crystals. We start with a detailed presentation of the elementary methods for the calculation of band structures and transmission phenomena which have proven their efficiency in the literature. We respectively present the plane wave expansion method, the finite difference time domain method, and the finite element method and discuss their respective advantages or drawbacks for different purposes and their applications to some current structures. The main objective is to give the elementary ingredients for graduated students to start the phononic crystal topic from the beginning. Beside we apply the methods to 2D infinite phononic crystals and show the condition of band gap existence as a function of the geometrical and physical parameters. Band gap maps of 2D phononic crystals have been presented by means of dispersion curve calculations. We then turn to the calculation of the transmission and present applications that can be easily reproduced, as the manipulation of acoustic waves through linear or point defect and to sensing applications. At lower frequencies, we present examples of low resonant sonic materials that were at the origin of the acoustic metamaterial (AMM) field introduced by P. Sheng in 2000. For AMMs, the driving wavelength is at least one order of magnitude higher than every dimension of the constitutive elements. We then present systematically the main physical properties of band gaps for different finite thickness 2D phononic crystal plates and discuss the condition of existence of the band gap for two kinds of phononic plates. The first one is a membrane drilled with air holes. The second one is made of pillar deposited on a thin membrane. The last structure opens the way to low frequency band gap that can be compared to low resonant sonic materials and now cover the field of AMMs.