The article "Inverse Problems: A Bayesian Perspective" by A. M. Stuart, published in Acta Numerica in May 2010, reviews the Bayesian approach to regularization in the context of inverse problems. The author emphasizes the importance of regularization to address ill-posedness and introduces a function space viewpoint that allows for a full characterization of all possible solutions and their relative probabilities. This approach, while computationally expensive, is becoming feasible in many application areas due to advancements in computational resources. It also enables the quantification of uncertainty and risk, which is increasingly demanded by practical applications. The article is divided into five parts. The first part provides an overview of the Bayesian framework, linking it to classical approaches and highlighting the role of observational noise. The second part discusses finite-dimensional inverse problems and the Bayesian solution, including the construction of the posterior measure. The third part explores the effect of small observational noise on the posterior measure, connecting the Bayesian and classical perspectives. The fourth part delves into the common mathematical structure underlying Bayesian inverse problems for functions, proving a form of well-posedness and an approximation theorem. The fifth part surveys algorithmic tools used to solve these problems, including Markov chain Monte Carlo (MCMC) methods, filtering methods, and variational methods. The author emphasizes the importance of avoiding discretization until the last possible moment, which is empowering in numerical analysis. This principle is applied to various problems, such as the first-order wave equation and the heat equation, to demonstrate its benefits. The article also discusses the application of Bayesian inverse problems in fields like atmospheric sciences, oceanography, geophysics, and molecular dynamics. The article concludes by highlighting the rich research interface between applied mathematics and statistics, where Bayesian methods offer a rigorous and transparent approach to inverse problems, making modeling assumptions explicit and addressing them explicitly.The article "Inverse Problems: A Bayesian Perspective" by A. M. Stuart, published in Acta Numerica in May 2010, reviews the Bayesian approach to regularization in the context of inverse problems. The author emphasizes the importance of regularization to address ill-posedness and introduces a function space viewpoint that allows for a full characterization of all possible solutions and their relative probabilities. This approach, while computationally expensive, is becoming feasible in many application areas due to advancements in computational resources. It also enables the quantification of uncertainty and risk, which is increasingly demanded by practical applications. The article is divided into five parts. The first part provides an overview of the Bayesian framework, linking it to classical approaches and highlighting the role of observational noise. The second part discusses finite-dimensional inverse problems and the Bayesian solution, including the construction of the posterior measure. The third part explores the effect of small observational noise on the posterior measure, connecting the Bayesian and classical perspectives. The fourth part delves into the common mathematical structure underlying Bayesian inverse problems for functions, proving a form of well-posedness and an approximation theorem. The fifth part surveys algorithmic tools used to solve these problems, including Markov chain Monte Carlo (MCMC) methods, filtering methods, and variational methods. The author emphasizes the importance of avoiding discretization until the last possible moment, which is empowering in numerical analysis. This principle is applied to various problems, such as the first-order wave equation and the heat equation, to demonstrate its benefits. The article also discusses the application of Bayesian inverse problems in fields like atmospheric sciences, oceanography, geophysics, and molecular dynamics. The article concludes by highlighting the rich research interface between applied mathematics and statistics, where Bayesian methods offer a rigorous and transparent approach to inverse problems, making modeling assumptions explicit and addressing them explicitly.