  Intermediate dimensions were recently introduced to interpolate between the
Hausdorff and box-counting dimensions of fractals. Firstly, we show that these
intermediate dimensions may be defined in terms of capacities with respect to
certain kernels. Then, relying on this, we show that the intermediate
dimensions of the projection of a set $E \subset \mathbb{R}^n$ onto almost all
$m$-dimensional subspaces depend only on $m$ and $E$, that is, they are almost
surely independent of the choice of subspace. Our approach is based on
`intermediate dimension profiles' that are expressed in terms of capacities. We
discuss several applications at the end of the paper, including a surprising
result that relates the box dimensions of the projections of a set to the
Hausdorff dimension of the set.
