The growth of computing power over recent decades has allowed scientists to implement rigorous computational techniques that had previously been unfeasible. Such was the approach adopted during the 'Realistic computational modelling of large-scale wave propagation problems in unbounded domains' (Wave propagation) project, which received EU funding. The focus of the research was on optimising the application of the scaled boundary finite element method (SBFEM), which essentially entails cutting the problem into smaller bits. Work to combine the SBFEM with the mixed-variables technique resulted in more accurate simulation of the propagation of elastic and acoustic waves in complex spaces. Building on previous research, a doubly asymptotic expansion of the SBFE dynamic stiffness was employed to address the mathematical challenge of accurately modelling radiation damping. Furthermore, a novel non-classical method of solving fractional differential equations was developed. This in turn enabled modelling of transient diffusion in a semi-infinite layered system directly in the time-domain, which had never before been accomplished. Instability is often responsible for rendering many computational solutions invalid. During the project, considerable progress was made in overcoming this obstacle in cases with a large number of degrees of freedom. The key was to modify the doubly asymptotic expansion based on a detailed analysis of the scalar wave equation formulated in spherical coordinates. This knowledge has been shared with the research community through several publications in peer-reviewed journals and presentations at relevant conferences.