A review of boundary element methods for the numerical solution of elastodynamic problems covering the period 1986-1996 is presented. It is a continuation of a review article on the same subject by the same author which appeared previously in Applied Mechanics Reviews (AMR 40 (1) 1-23 (Jan 1987) Reprint No AMR015). Integral formulations and their advanced numerical treatment in both frequency and time domains from the direct boundary element method viewpoint are described. They cover two - and three - dimensional cases as well as the anti-plane case of linear elastodynamics under harmonic or transient disturbances. Indirect formulations, boundary methods, T-matrix methods, symmetric formulations, dual reciprocity boundary element methods and hybrid schemes combining boundary with finite elements are also described. All these boundary element methodologies are applied to: i) wave propagation analysis including wave propagation due to external loads, wave diffraction by surface or subsurface irregularities and cracks and crack propagation; ii) dynamic analysis of structures including beams, membranes, plates and shells as well as two - and three - dimensional structures; iii) soil-structure interaction including foundation analysis, piles and underground structures; iv) fluid-structure interaction including structures inside fluids or containing fluids and dam-reservoir systems; and v) the special subjects of viscoelasticity, inhomogeneity, anisotropy, poroelasticity-thermoelasticity, large deformations, contact analysis, inverse scattering and optimum design and control. Finally, areas where further research is needed are identified. There are 1333 references.