Discontinua simulations are becoming an important part of computational mechanics to the extent that computational mechanics of discontinua is becoming a separate sub–discipline of computational mechanics. Among the most widely used methods of computational mechanics of discontinua are discrete–element methods, combined finite–discrete–element methods and discontinuum deformation analysis methods. A range of key algorithmic procedures is common to most of these methods. These include contact detection, explicit solvers, fracture and fragmentation models, handling of complex geometric considerations when processing interaction in three dimensions (contact kinematics) and fluid coupling. In recent years, there have been major breakthroughs in almost all of these key algorithmic aspects. These include linear contact–detection procedures (NBS, C–grid), discretized contact solutions, fracture and fragmentation solutions, together with fluid pressure driven fracture process and three–dimensional explicit solvers incorporating finite rotations. Many of these breakthroughs have not yet been applied across the full range of relevant discontinuum problems. The major reason for this is that discrete–element method, discontinuum deformation analysis and combined finite–discrete–element method publications are spread over a wide range of specialist journals and conferences. Thus in this paper, the main features of a selection of algorithmic breakthroughs are reviewed for the first time, enabling researchers in different fields to apply these compatible developments to their specific applications.