Uncertainty quantification approaches are generally computationally intractable for large-scale complex systems. The discretized forward models describing such systems typically are of very high dimension and are expensive to solve. The computational resources required for uncertainty quantification therefore quickly become prohibitive. Model reduction can address this challenge by producing low-order approximate models that retain the essential system dynamics but that are fast to solve. This talk will discuss formulations of model reduction problems for applications in uncertainty quantification. Key challenges include systems with input parameter spaces of very high dimension (infinite-dimensional parameters in some cases), and accounting for the statistical properties of interest in the system outputs. We demonstrate the use of reduced models for uncertainty propagation, solution of statistical inverse problems, and optimization under uncertainty for systems governed by partial differential equations. Our methods use state approximations through the proper orthogonal decomposition, reductions in parameter dimensionality through parameter basis approximations, and the empirical interpolation method for efficient evaluation of nonlinear terms.