These articles, corresponding to talks given at the Foundations of Computational Mathematics conference, bring together a variety of subjects in which the computational process itself and the foundational mathematical underpinnings of algorithms are the objects of study. The conference was organized around nine workshops - systems of algebraic equations and computational algebraic geometry, homotopy methods and real machines, information-based complexity, numerical linear algebra, approximation and PDEs, optimization, differential equations and dynamical systems, relations to computer science and vision, and related computational tools.