Explore recent developments in regularity theory for nonlinear PDEs in this 49-minute lecture by Edgard Pimentel, presented as part of the Hausdorff Trimester Program on Evolution of Interfaces. Delve into a two-layered approach that connects complex problems to those with established theories, utilizing compactness and stability concepts for approximation, followed by scaling arguments for localized analysis. Examine toy models including fully nonlinear PDEs, the Isaacs equation, double-divergence problems, and degenerate/singular equations. Progress through topics such as existence of small correctors, oscillation control, non-variational settings, Hölder regularity of gradients, and approximation lemmas. Conclude with an exploration of free boundary problems, gaining insights into sharp regularity results and analytical developments in this field.