Masonry structures are represented by discrete element models as assemblies of blocks or particles, an idealization of their discontinuous nature that governs the mechanical behavior. This ‘discontinuum’ approach, an alternative to modeling masonry as a homogenized continuum, is particularly suited for detailed models used in fundamental research and the interpretation of experiments. Computational developments have allowed these techniques to be progressively applied in engineering practice, namely in the failure analysis of structural components and larger structures, under static or earthquake loading. A review is presented of the main models based on the discrete element and related numerical techniques that have been proposed for the analysis of masonry. The essential assumptions adopted by these models and numerical implementation issues are discussed. Differences between available models are illustrated by applications to various masonry problems that are very effectively analyzed by discrete elements.