Luis A. Caffarelli earned his Ph.D. in mathematics from the University of Buenos Aires. He is now professor of mathematics and holds the Sid W. Richardson Foundation Regents’ Chair in Mathematics No. 1 at the University of Texas at Austin. He is also a core member of Institute for Computational Engineering and Sciences there. The focus of Professor Caffarelli’s research has been in the area of elliptic nonlinear partial differential equations and their applications. His research has reached from theoretical questions about the regularity of solutions to fully nonlinear elliptic equations to partial regularity properties of Navier Stokes equations. Some of his most significant contributions are the regularity of free boundary problems and solutions to nonlinear elliptic partial differential equations, optimal transportation theory and, more recently, results in the theory of homogenization. In a series of papers starting in 1990, Caffarelli studied viscosity solutions to non- linear partial differential equations, both the Monge–Ampère equation and the equation that models flow in a porous medium. This has proven to be an important means to arrive at the existence and uniqueness of solutions. As a result, Caffarelli has been cited as the world’s leading specialist in free-boundary problems for nonlinear partial differential equations, and a pioneer in methods tackling many classical problems that have long defied mathematicians. With his collaborators, he has authored more than 280 scientific publications documenting this work. Caffarelli has received numerous honors and awards including seven honorary doctorates, the Stampacchia Medal from the Italian Mathematical Union, the Bocher Memorial Prize from the American Mathematical Society, the Pius XI Gold Medal from the Pontifical Academy of Sciences, the Premio Konex, Platino y Brillantes from the Konex Foundation in Argentina, and the Rolf Schock Prize from the Swedish Royal Academy of Sciences. He also received the Leroy P. Steele Prize for Lifetime Achievement in Mathematics from the American Mathematical Society. In 2012 he received Israel’s Wolf Prize in Mathematics. The following year he was honored with the first ever, Solomon Lefschetz Medal at the inaugural Mathematical Congress of the Americas. ABSTRACT: Non local diffusion problems arise in many areas of mathematics: phase transitions, constrained configurations, quasilinear equation types, financial engineering. I will present some of the problems and discuss the mathematical ideas involved in their study.