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We first introduce the class of R–cubings, which specialise the class of complete geodesic median metric spaces of finite rank in roughly the same way that hierarchically hyperbolic spaces specialise coarse median spaces. The first of our main results says that, if X is a hierarchically hyperbolic space, then any asymptotic cone of X is bilipschitz equivalent to an R–cubing. This generalises the fact that asymptotic cones of hyperbolic groups are R–trees, and also generalises and strengthens a result of Behrstock-Druțu-Sapir on asymptotic cones of mapping class groups. This makes essential use of a result of Bowditch about medians on asymptotic cones of coarse median spaces, as well as Fioravanti’s work on measured halfspaces. We then introduce the notion of a universal R–cubing, which is a homogeneous R–cubing whose structure is completely determined by the local R–cubing structure at any point. We show that asymptotic cones of G are bilipschitz equivalent to universal R–cubings. This reduces the problem of studying asymptotic cones of G to that of understanding the local structure. Under algebraic conditions on a hierarchically hyperbolic group G—satisfied by the motivating examples of mapping class groups of surfaces, fundamental groups of compact special cube complexes, and hyperbolic groups—, we prove that (up to bilipschitz equivalence), the asymptotic cones of G are independent of the ultrafilter and rescaling sequence.