Operator learning involves using machine learning techniques to approximate nonlinear operators between Banach spaces of functions, often arising from physical models described by partial differential equations (PDEs). These operators can serve as efficient surrogate models for traditional numerical methods and enable model discovery when a mathematical description is unavailable. This review focuses on neural operators, which build on the success of deep neural networks in approximating functions on finite-dimensional spaces. Empirically, neural operators have shown promise in various applications, but theoretical understanding remains incomplete. The paper summarizes recent progress and theoretical insights into neural operators, focusing on approximation theory. The paper discusses the motivation for operator learning, reviews existing literature, and outlines the remainder of the paper. It emphasizes the importance of viewing high-dimensional vectors as functions, which allows for more intrinsic problem representation. The paper then reviews algorithms for function space learning, including supervised learning on function spaces, and discusses various neural operator architectures such as PCA-Net, DeepONet, Fourier Neural Operators (FNO), and random features. It also covers the theoretical foundations of universal approximation for these architectures, demonstrating their ability to approximate a wide range of operators. The paper concludes with an overview of the paper's structure and the key findings in operator learning.Operator learning involves using machine learning techniques to approximate nonlinear operators between Banach spaces of functions, often arising from physical models described by partial differential equations (PDEs). These operators can serve as efficient surrogate models for traditional numerical methods and enable model discovery when a mathematical description is unavailable. This review focuses on neural operators, which build on the success of deep neural networks in approximating functions on finite-dimensional spaces. Empirically, neural operators have shown promise in various applications, but theoretical understanding remains incomplete. The paper summarizes recent progress and theoretical insights into neural operators, focusing on approximation theory. The paper discusses the motivation for operator learning, reviews existing literature, and outlines the remainder of the paper. It emphasizes the importance of viewing high-dimensional vectors as functions, which allows for more intrinsic problem representation. The paper then reviews algorithms for function space learning, including supervised learning on function spaces, and discusses various neural operator architectures such as PCA-Net, DeepONet, Fourier Neural Operators (FNO), and random features. It also covers the theoretical foundations of universal approximation for these architectures, demonstrating their ability to approximate a wide range of operators. The paper concludes with an overview of the paper's structure and the key findings in operator learning.