  The first main result of this paper is that the law of the (rescaled)
two-dimensional uniform spanning tree is tight in a space whose elements are
measured, rooted real trees continuously embedded into Euclidean space. Various
properties of the intrinsic metrics, measures and embeddings of the
subsequential limits in this space are obtained, with it being proved in
particular that the Hausdorff dimension of any limit in its intrinsic metric is
almost surely equal to $8/5$. In addition, the tightness result is applied to
deduce that the annealed law of the simple random walk on the two-dimensional
uniform spanning tree is tight under a suitable rescaling. For the limiting
processes, which are diffusions on random real trees embedded into Euclidean
space, detailed transition density estimates are derived.
