The uniform spanning forests (USFs) of infinite an infinite graph G are defined as infinite volume limits of uniform spanning trees on finite subgraphs of G. In this talk, I will describe how we use a new way of sampling the USF using the random interlacement process to compute various critical exponents governing the large-scale geometry of trees in the forest in a wide variety of “high-dimensional” graphs, including Z^d for d \geq 5 and every bounded degree nonamenable graph. I will then sketch how this allows us to compute related exponents describing the geometry of avalanches in the Abelian sandpile model on the same class of graphs.