This textbook focuses on computational methods for inverse problems that are governed by partial differential equations (PDEs). The author considers deterministic and Bayesian formulations and highlights how traditional tools from deterministic inversion can be integrated into solution methods for Bayesian inverse problems. Advanced topics such as post-optimality sensitivity analysis, optimal design of experiments, and Bayesian inversion under model uncertainty are also included. Computational Inverse Problems Governed by PDEs offers readers a balance of theoretical and computational insight, an example-driven approach that provides an accessible presentation, and over 150 theoretical and computational exercises.