Exponential integrators are numerical methods for solving stiff systems of differential equations, particularly those with large negative real parts in their Jacobian or with purely imaginary eigenvalues of large modulus. This paper discusses the construction, analysis, implementation, and application of exponential integrators. The focus is on two types of stiff problems: parabolic partial differential equations and highly oscillatory problems. The paper presents the mathematics behind these methods, deriving error bounds that are independent of stiffness or highest frequencies in the system. It also discusses implementation issues, such as evaluating matrix functions efficiently, and provides applications in science and technology. The paper concludes with historical remarks and references. The main idea is to use the exact solution of a prototypical equation to construct numerical methods, leading to exponential integrators. The paper also discusses the convergence properties of exponential integrators for finite times and their application to various types of problems, including parabolic problems and highly oscillatory problems. The paper highlights the importance of the stiff order in determining the convergence of exponential integrators and provides examples of exponential Runge–Kutta and Rosenbrock methods. The paper also discusses the implementation of exponential integrators, including the use of Krylov subspace methods and contour integral methods. The paper concludes with a discussion of the historical development of exponential integrators and their applications in various fields.Exponential integrators are numerical methods for solving stiff systems of differential equations, particularly those with large negative real parts in their Jacobian or with purely imaginary eigenvalues of large modulus. This paper discusses the construction, analysis, implementation, and application of exponential integrators. The focus is on two types of stiff problems: parabolic partial differential equations and highly oscillatory problems. The paper presents the mathematics behind these methods, deriving error bounds that are independent of stiffness or highest frequencies in the system. It also discusses implementation issues, such as evaluating matrix functions efficiently, and provides applications in science and technology. The paper concludes with historical remarks and references. The main idea is to use the exact solution of a prototypical equation to construct numerical methods, leading to exponential integrators. The paper also discusses the convergence properties of exponential integrators for finite times and their application to various types of problems, including parabolic problems and highly oscillatory problems. The paper highlights the importance of the stiff order in determining the convergence of exponential integrators and provides examples of exponential Runge–Kutta and Rosenbrock methods. The paper also discusses the implementation of exponential integrators, including the use of Krylov subspace methods and contour integral methods. The paper concludes with a discussion of the historical development of exponential integrators and their applications in various fields.