Quantifying uncertainties in systems governed by partial differential equations is an important endeavor in simulation, design, and control in all areas of science and engineering. Monte Carlo methods are in very common use but are limited by their high cost. Alternatives methods are discussed that reduce costs by lowering the number and/or cost of the PDE solves used. The course is self contained: short presentations about PDEs, their finite element approximation, and notions from probability are discussed.