One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows:  for a wide array of random graph models  on  n vertices and degree exponent \tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}. In other words, the degree exponent determines the universality class the random graph belongs to. More generally, recent research has provided strong evidence to believe that several objects constructed on a wide class of random discrete structures including(a) components under critical percolation,(b) the vacant set left by a random walk, and(c) the minimal spanning tree,viewed as metric measure spaces converge, after scaling the graph distance, to some random fractals, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures.Based on joint work with Shankar Bhamidi, Nicolas Broutin, Remco van der Hofstad, and Xuan Wang.