For E a subset of ℝn and 0 [les ] m [les ] n we define a ‘family of dimensions’ DimmE, closely related to the packing dimension of E, with the property that the orthogonal projection of E onto almost all m-dimensional subspaces has packing dimension DimmE. In particular the packing dimension of almost all such projections must be equal. We obtain similar results for the packing dimension of the projections of measures. We are led to think of DimmE for m ∈ [0, n] as a ‘dimension profile’ that reflects a variety of geometrical properties of E, and we characterize the dimension profiles that are obtainable in this way.