This article presents a Bayesian perspective on inverse problems, focusing on their formulation and solution in function spaces. It discusses the importance of regularization in addressing ill-posedness and highlights the advantages of the Bayesian approach in quantifying uncertainty and risk. The article reviews the common mathematical structure underlying various inverse problems and introduces a framework for analyzing them in infinite-dimensional settings. It also surveys algorithmic approaches such as Markov chain Monte Carlo (MCMC), filtering, and variational methods. The Bayesian framework is shown to provide a coherent and flexible approach to inverse problems, allowing for the incorporation of prior knowledge and the quantification of uncertainty. The article emphasizes the importance of avoiding discretization until the final stages of algorithmic formulation, which is crucial for maintaining the accuracy and efficiency of numerical methods. It also discusses the connection between Bayesian and classical approaches to inverse problems, showing how the Bayesian perspective can lead to a deeper understanding of the underlying mathematical structure. The article concludes with a discussion of the role of the prior in the solution of inverse problems and the importance of careful modeling in the Bayesian framework.This article presents a Bayesian perspective on inverse problems, focusing on their formulation and solution in function spaces. It discusses the importance of regularization in addressing ill-posedness and highlights the advantages of the Bayesian approach in quantifying uncertainty and risk. The article reviews the common mathematical structure underlying various inverse problems and introduces a framework for analyzing them in infinite-dimensional settings. It also surveys algorithmic approaches such as Markov chain Monte Carlo (MCMC), filtering, and variational methods. The Bayesian framework is shown to provide a coherent and flexible approach to inverse problems, allowing for the incorporation of prior knowledge and the quantification of uncertainty. The article emphasizes the importance of avoiding discretization until the final stages of algorithmic formulation, which is crucial for maintaining the accuracy and efficiency of numerical methods. It also discusses the connection between Bayesian and classical approaches to inverse problems, showing how the Bayesian perspective can lead to a deeper understanding of the underlying mathematical structure. The article concludes with a discussion of the role of the prior in the solution of inverse problems and the importance of careful modeling in the Bayesian framework.