Given a solution of a nonlinear pde the two primary issues regarding the regularity theory are a priori estimates and the structure of the singular set. We will discuss new techniques in the analysis of these issues, which have been particularly successful in the study of geometrically motivated equations. To pick an example: in the context of a stationary harmonic map f:M->N between Riemannian manifolds we will see that the singular set S(f) may be stratified into pieces S^k(M) which are k-rectifiable. If f is minimizing harmonic map then we will see that the singular set S(f) has finite n-3 measure, and has apriori estimates in weak L^3 , both of which are sharp estimates. The techniques of the proofs are quite general and include the quantitative stratification, W^{1,p}-Reifenberg results, and L^2 -subspace approximation results, and also work to give sharp estimates for minimal surfaces and other geometric equations. We will try and give an understandable introduction to these topics. This is joint work with Daniele Valtorta.