Inverse problems concern inferring unknown parameters and/or inputs of a system from partial observations. This chapter provides a high-level overview of the field, with an emphasis on how to pose inverse problems as an optimization problem. Most inverse problems are fundamentally ill-posed, and so they involve regularization, often in the form of physics and priors. Control theory and machine learning are both inverse problems. There are many other examples of inverse problems in the modern world. Two of the most compelling and relatable examples are in medical imaging and computational cosmology. In this chapter, we will cover these classical applications of inverse problems, including the Radon and Fourier transforms. We will also explore modern problems in image processing, PDE-constrained inverse problems, and system identification.