The modelling of unbounded domains is an important consideration in many engineering problems, for example in fluid flow, electro-magnetics, acoustics and solid mechanics. This thesis focuses on the problem of modelling elastic solids to infinity, with the specific purpose of modelling dynamic soil-structure interaction (DSSI). However, the reader should be aware that the techniques presented may also be adapted to address those other physical phenomena. The need for techniques to model the soil domain to infinity and a qualitative introduction into the problems associated with dynamic soil-structure interaction are outlined in chapter 1. This is done to illustrate why such an abstract mathematical concept of modelling infinite domains has an important role to play within the design process of large, safety critical, civil engineering structures. A brief review of a number of alternative ways of addressing this problem is given in chapter 2. Their relative strengths and weaknesses along with the typical applicability of the techniques is discussed. A consequence of this review is the identification of a very promising rigorous approach [59] which is singled-out for further study. A detailed explanation of this (Consistent Infinitesimal Finite Element Cell Method, CIFECM) method is then given in chapter 3. Attention is restricted to the use of the technique for solving the 3-D vector wave equation in the time domain. The features of the non-linear dynamic finite element code, into which the CIFECM has been incorporated, is highlighted in chapter 4. The non-linear (microplane) material model for quasi-brittle materials is described along with the solution strategy employed. It should be mentioned that the soil is treated within this thesis as drained linear elastic medium. The method of coupling the CIFECM into the dynamic equation of force equilibrium for both directly applied and transmitted loading regimes is detailed. Application of the code follows in chapter 5; firstly by introducing the simplest test problem of one finite element coupled with one CIFECM element to model a surface foundation. Comparisons are made between the dynamic displacements resulting from the method and standard FE solutions obtained from the use of extended meshes and fixed boundary conditions, along with a study of the influence input variables. Following these examples a larger (more realistic) engineering problem is tacked involving the simulation of an aircraft impact on a reinforced concrete nuclear containment vessel. This represents the first use of the method in a 3-D nonlinear structural analysis problem. The results illustrate the practical implications of including DSSI in the analysis. III In chapter 6, a series of general observations on the method are made with an assessment of its value together with a discussion on its wider application to other engineering fields. Possible future developments to make the method more computationally efficient are finally suggested.