  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 L\'{e}ger's
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 $\beta$-numbers in terms of integral Menger
curvature without assuming lower Ahlfors regularity, in contrast to the results
of Lerman and Whitehouse.
