The main purpose of this work is to develop methods capable of modelling the discontinuous behaviour of un-reinforced structural masonry with a reasonable degree of accuracy and to demonstrate these methods and their applicability. The modelling frameworks that are currently being used for the analysis of structural masonry are reviewed. The principles of modelling at micro, macro and simplified-micro scales are discussed. Detailed micro-models and their applicability to small samples and test specimens of masonry are discussed. Macro-models and homogenisation techniques and their applicability to large uniform expanses of masonry are discussed. Simplified micro-models in their continuum- and discontinuum-based forms are presented as a suitable compromise. Discontinuous Deformation Analysis (DDA) is presented as a suitable modelling framework. This is an implicit method based upon the solution of the equations of motion for a discontinuous blocky system. The equations of motion are derived by minimisation of potential energy. Efficient methods of solving the equilibrium equations are explored. A technique involving a mapping of DDA blocks into finite elements is presented as a flexible approach to increase the accuracy of the deformability of discrete bodies in a structure. This method is termed DDA-FE. The implicit DDA contact mechanics routines are retained in the DDA-FE framework. Tree based and grid-search based contact detection techniques are discussed and a grid-search binning algorithm is implemented for the DDA-FE framework. Methods of enforcing contact constraints are discussed including the local contact resolution techniques. An implicit contact predictor-corrector method of enforcing Mohr-Coulomb functional contact constraints is presented. The DDA-FE framework that has been developed is verified and validated against a series of illustrative examples and benchmark problems. A detailed analysis of the Couplet-Heyman minimum thickness arch problem is conducted. The implicit format of the DDA-FE framework is exploited to show that an eigenvalue analysis of the structural stiffriess can be a valuable tool for assessing the stability of a discontinuous masonry structure. It has also been shown that an eigenvalue analysis can be used to predict the failure modes of discontinuous structures whose behaviour is non-smooth and difficult to predict by continuum based methods. Material non-linearity is introduced to the DDA-FE framework. Two pre-fracture non-linear continuum models are implemented, a bi-linear-elastic model and an elastic-perfectly-plastic model, both based upon mean element stress. A discrete fracture model is implemented to model fragmentation. These three models are verified using a series of illustrative examples. The developed methods are used successfully to model in detail the Bridgemill masonry arch, a real world structure that was tested to destruction in order to provide data about the behaviour of such structures.