In this thesis, we use plane wave expansion method (PWEM) and multiple scattering method to study the propagation behavior of elastic waves in the two-dimensional phononic crystals. The band structure of mixed (longitudinal-transverse) polarization mode and transverse polarization mode can be calculated using PWEM. The band structure for longitudinal polarization mode of two-dimensional phononic crystal composed of non-viscous fluid, and the band structure of transverse polarization mode of two-dimensional phononic crystal of solid can also be calculated. Inspired by the research of dielectric rods waveguide for guiding light, we propose elastic rods waveguide for guiding sound. The band structure of elastic rods waveguide is calculated using super-cell method, and the propagating mode can be obtained. We also consider the effect of bends and disorder on the transmission of the elastic rods waveguide. We believe that this kind of acoustic waveguide can useful in the future.