This thesis provides a rigorous framework for the solution of stochastic elliptic partial differential equation (SPDE) constrained optimization problems. In modeling physical processes with differential equations, much of the input data is uncertain (e.g. measurement errors in the diffusivity coefficients). When uncertainty is present, the governing equations become a family of equations indexed by a stochastic variable. Since solutions of these SPDEs enter the objective function, the objective function usually involves statistical moments. These optimization problems governed by SPDEs are posed as a particular class of optimization problems in Banach spaces. This thesis discusses Monte Carlo, stochastic Galerkin, and stochastic collocation methods for the numerical solution of SPDEs and identifies the stochastic collocation method as particularly useful for the optimization of SPDEs. This thesis extends the stochastic collocation method to the optimization context and explores the decoupling nature of this method for gradient and Hessian computations.