The purpose of this lecture is to introduce the main tools for studying the gravitational N-body problem by the direct summation approach. Although the emphasis will be on star cluster simulations, some methods are also suitable for planetary problems. Algorithms for workstations, laptops and the GRAPE special-purpose supercomputer will be presented. We first describe the basic Hermite integration with neighbour scheme and the treatment of close encounters by two-body regularization. Procedures for consistent modelling of stellar evolution effects in star clusters are reviewed. A brief discussion is also given of measures for dealing with strong interactions in multiple systems as well as the stability of hierarchies. Finally, we outline an implementation on the GRAPE with particular application to the binary black hole problem, including post-Newtonian terms.
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