  We show that the Brownian continuum random tree is the
Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on
high-dimensional graphs including the $d$-dimensional torus $\mathbb{Z}_n^d$
with $d>4$, the hypercube $\{0,1\}^n$, and transitive expander graphs. Several
corollaries for associated quantities are then deduced: convergence in
distribution of the rescaled diameter, height and simple random walk on these
uniform spanning trees to their continuum analogues on the continuum random
tree.
