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We develop a general universality technique for establishing metric scalinglimits of critical random discrete structures exhibiting mean-field behaviorthat requires four ingredients: (i) from the barely subcritical regime to thecritical window, components merge approximately like the multiplicativecoalescent, (ii) asymptotics of the susceptibility functions are the same asthat of the Erdos-Renyi random graph, (iii) asymptotic negligibility of themaximal component size and the diameter in the barely subcritical regime, and(iv) macroscopic averaging of distances between vertices in the barelysubcritical regime.  As an application of the general universality theorem, we establish, undersome regularity conditions, the critical percolation scaling limit of graphsthat converge, in a suitable topology, to an $L^3$ graphon. In particular, wedefine a notion of the critical window in this setting. The $L^3$ assumptionensures that the model is in the Erdos-Renyi universality class and that thescaling limit is Brownian. Our results do not assume any specific functionalform for the graphon. As a consequence of our results on graphons, we obtainthe metric scaling limit for Aldous-Pittel's RGIV model [9] inside the criticalwindow.  Our universality principle has applications in a number of other problemsincluding in the study of noise sensitivity of critical random graphs [52]. In[10], we use our universality theorem to establish the metric scaling limit ofcritical bounded size rules. Our method should yield the critical metricscaling limit of Rucinski and Wormald's random graph process with degreerestrictions [56] provided an additional technical condition about the barelysubcritical behavior of this model can be proved.

Scaling limits and universality: Critical percolation on weighted graphs converging to an $L^3$ graphon