In this work the direct boundary element method is applied to solve transient wave propagation problems. At first, the scalar wave equation is considered and the discussion initially carried out illustrates the mathematical operations that are required in order to obtain two- and three-dimensional boundary integral equations which are amenable to numerical solutions. Linear discretization is adopted to represent the boundary geometry with linear and constant time and space interpolation functions being employed to approximate the boundary unknowns. Consequently the two-dimensional boundary integral equation is transformed into a system of algebraic equations, which is solved by implementing a time-stepping scheme in which time integrations are carried out analytically. One-dimensional Gauss quadrature is used to perform all boundary integrals except those in the Cauchy principal value sense which are calculated analytically. Linear triangular cells are used to compute contributions due to initial conditions. An investigation concerning elastodynamics, where two- and three-dimensional formulations are considered is also included. The numerical procedure which is employed in solving two-dimensional elastodynamic problems is very similar to that concerning the scalar wave equation, for this reason the discussion concerning this subject is only cursory. Initial conditions are not included, but cells are also used in the elastodynamic analysis, to compute internal stresses. A number of examples which relate to the two wave propagation problems previously mentioned are analysed and numerical results together with discussions regarding their accuracy are included. Certain other topics are also considered like the number of integration points that should be used, the relation between the element length and time interval size that should be chosen, etc.