Abstract. In this work we show that an n-dimensional Borel set in Euclidean N -space with finite integral Menger curvature is n-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null set in the sense of Hausdorff measure. This generalises Léger's [19] rectifiability result for one-dimensional sets to arbitrary dimension and co-dimension. In addition, we characterise possible integrands and discuss examples known from the literature.Intermediate results of independent interest include upper bounds of different versions of P. Jones's β-numbers in terms of integral Menger curvature without assuming lower Ahlfors regularity, in contrast to the results of Lerman and Whitehouse [20].

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