In this series of talks, we cover the current finite difference, finite element, and 
discontinuous Galerkin methods used to approximate the solutions of second-order elliptic 
fully nonlinear PDEs.  In part one, we introduce examples of some simple fully nonlinear 
PDEs and outline the solution theory of these equations.  Classical solutions of fully 
nonlinear PDEs rarely exist and thus we cover the viscosity solution concept used to solve 
them.  Existence and uniqueness theory for viscosity solutions is summarized using 
Perron's method and the vanishing viscosity method.  We finish by mentioning two second-
order equations which will be our focus for the remainder of the series: the Hamilton-
Jacobi-Bellman equation and the Monge-Ampère equation.