Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex polyhedron P is rigid i.e. every continuous motion of the vertices of P in R^3 which preserves its edge lengths results in a polyhedron which is congruent to P. This result was extended to convex poytopes in R^d for all d\geq 3 by Whiteley, and to generic realisations of 1-skeletons of simplicial (d-1)-manifolds in R^d by Kalai for d\geq 4 and Fogelsanger for d\geq 3. We will generalise Kalai's result by showing that, for all \geq 4 and any fixed 1\leq k\leq d-3, every generic realisation p of the k-skeleton of a simplicial (d-1)-manifold S in R^d is volume rigid, i.e. every continuous motion of the vertices of (S,p) in R^d which preserves the volumes of its k-faces results in a realisation which is congruent to p. In addition, we conjecture that our result remains true for k=d-2 and verify this conjecture when d=4,5. This is joint work with James Cruickshank and Shin-Ichi Tanigawa.