A spanning tree of a finite connected graph $G$ is a connected subgraph of $G$ that touches every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees of high-dimensional graphs, and explain why, under appropriate rescaling, they converge in distribution as metric-measure spaces to Aldous' Brownian CRT. This extends an earlier result of Peres and Revelle (2004) who previously showed a form of finite-dimensional convergence. Based on joint works with Asaf Nachmias and Matan Shalev.