The dominated hypervolume (also known as Lebesgue measure or S-metric ) maps a set of points in objective space to the size of of the region Pareto dominated by that set. The hypervolume has to be bounded from below by a reference point, which for the purposes of this paper will systematically be chosen to be the origin of objective space. The dominated hypervolume simultaneously takes into account the proximity of the points to the ideal Pareto front and their spread over the objective space. For problems with less than five objectives, the dominated hypervolume can be calculated exactly. However, for higher-dimensional multi-objective optimisation problems, calculating the dominated hypervolume precisely can be computationally expensive and hence a smorgasbord of efficient approximation methods for the dominated hypervolume has been developed.