The aim of this book is to serve as an introduction to both the theory of timedomain boundary integral equations (TDBIE) and the analysis of their numerical discretisation by convolution quadrature and the Galerkin boundary element method (BEM). This includes describing the main algorithms, giving an indication of the acceleration possible by the use of data sparse methods, and throughout describing some experiences of the authors from years of working in this area and giving some practical advice. The emphasis of the book is on hyperbolic problems, specifically acoustic scattering and to a much lesser extent scattering of electromagnetic waves. Due to some recent interest of the first author, we have also included a chapter on fast methods for parabolic problems. One of the main motivations behind writing this book has been to go beyond the standard sound–soft and sound–hard examples of scattering by obstacles. Therefore, we also describe applications involving nonlinear absorbing boundary conditions and FEM/BEM coupling. Another important motivation was to provide short Matlab codes that implement the main algorithms and illustrate the numerical performance of the schemes. The codes can be downloaded from the dedicated GitHub page https://github. com/lehelb/TDBIE-CQ-book where translations into Python and further codes and examples can be found. Errata, when they come as they surely will, will also be posted there. While this is envisioned as a graduate level text, the prerequisites for fully understanding the text are not few. Nevertheless, most of the background required should be part of a solid undergraduate+ master level mathematical education, and the appendices should serve to refresh this knowledge. One clear exception is the theory of steady-state boundary integral equations and their numerical discretisation by the boundary element method. Giving an introduction to this complex theory and practical implementation would have changed the nature of this book (and its length!). Fortunately, a number of excellent textbooks are now available in the literature that we refer to throughout our text. As for other books on numerical methods for TDBIE, let us first mention the earlier book by the second author [104]. While there are naturally many vii