Band gap materials(i.e.phononic crystals) are the artificially periodic structures,which have the stop band characteristic for elastic waves.The elastic waves will be localized in phononic crystals with defects,which results in the energy being accumulated around the defects.As a result,it is important to analyze the wave propagation and localization in band gap materials,especially for the structures consisting of smart materials.For example,with the mechanical-electro and mechanical-electro-magneto coupling,the phononic crystals consisting of piezoelectric and magnetoelectroelastic materials can be applied widely.This sets the theoretical basis for the design of band gap materials with multi fields coupling.This paper reviews the recent development of the elastic wave propagation and localization in both ordered and disordered band gap materials.The discussion focuses on the stop band and localization characteristics of elastic waves.Analytical methods and important results are also presented.Finally,some problems for further studies are discussed.This work aims to present the basic properties of wave band gaps in phononic crystals and wave localization in disordered periodic structures(e.g.phononic crystals with definite and random defects and phononic quasicrystals). Band gap materials (iephononic crystals) are the artificially periodic structures, which have the stop band characteristic for elastic waves.The elastic waves will be localized in phononic crystals with defects, which results in the energy is accumulated around the defects. As a result , it is important to analyze the wave propagation and localization in band gap materials, especially for the structures consisting of smart materials. For example, with the mechanical-electro and mechanical-electro-magneto coupling, the phononic crystals consisting of piezoelectric and magnetoelectroelastic materials can be applied widely. This sets the theoretical basis for the design of band gap materials with multi-field coupling. This paper reviews the recent development of the elastic wave propagation and localization in both ordered and disordered band gap materials. discussion focus on the stop band and localization characteristics of elastic waves.Analytical methods and important results are also presented.Finally, some problems for further studies are discussed. This work aims to present the basic properties of wave band gaps in phononic crystals and wave localization in disordered periodic structures (e.g. phononic crystals with definite and random defects and phononic quasicrystals).