This thesis considers the treatment of the wave equation given outside of a bounded, orientable Lipschitz domain with the boundary element method (BEM). Beginning with a scattering problem the retarded (potential) boundary integral operators are defined. These operators are discretized with a tensor product ansatz. For the retarded Poincar\'{e}-Steklov operator and the inverse counterpart, numerical experiments are presented using the marching-on-in time (MOT) scheme.The coupling of the finite element method (FEM) and the boundary element method (BEM) provide an analysis of a fluid-structure interaction (FSI) problem with given transmission conditions and the wave propagation interface problem with corresponding transmission conditions. For the FSI problem two approaches are addressed.The symmetric FEM-BEM coupling are discretized such that the MOT-scheme is applicable. Numerical experiments demonstrate the reliability of the implementation. The other approach uses a retarded boundary integral operator as a test function, which leads to major challenges in the discretization and the performing of numerical experiments. The wave propagation interface problem is adressed with a symmetric coupling. Here the discretization is chosen such that a MOT-scheme may applied. Numerical results are demonstrated as well. A prori and a posteriori error estimates for conforming Galerkin approximation are derived in all these cases, motivating adaptive mesh refinement procedures.The remaining chapters consider the results of time domain boundary element discretizations for screen problems, unilateral contact and a real-world application on tyres. Numerical experiments achieve optimal approximation rates on graded meshes for screen problems, resolving the edge and corner singularities. As a first step towards high-order methods $p$ and $hp-$versions of time domain boundary element method are presented for quasi-uniform meshes. Further crack and punch problems, as two examples of dynamic contact problems in time domain, are analyzed. While an error analysis is done for flat contact areas, numerical experiments show convergence even for non-flat contact areas. The sound emission of tyres, where noise emitting from the contact of the tyre with the pavement, are discussed. Numerical experiments illustrate the applicability of the boundary element method to real-world problems.