This paper focuses on the construction, analysis, implementation, and application of exponential integrators for solving stiff systems of differential equations. The authors consider two main types of stiff problems: those with a Jacobian that has eigenvalues with large negative real parts, typical for parabolic partial differential equations, and those with highly oscillatory solutions characterized by purely imaginary eigenvalues of large modulus. The primary goal is to derive error bounds that are independent of stiffness or the highest frequencies in the system. The paper discusses the mathematics behind exponential integrators, including the linearization process and the use of exact solutions to construct numerical schemes. It also covers the implementation of matrix function evaluations and provides an overview of applications in various fields such as quantum dynamics, chemistry, and mathematical finance. The authors conclude with a historical perspective on the development of exponential integrators.This paper focuses on the construction, analysis, implementation, and application of exponential integrators for solving stiff systems of differential equations. The authors consider two main types of stiff problems: those with a Jacobian that has eigenvalues with large negative real parts, typical for parabolic partial differential equations, and those with highly oscillatory solutions characterized by purely imaginary eigenvalues of large modulus. The primary goal is to derive error bounds that are independent of stiffness or the highest frequencies in the system. The paper discusses the mathematics behind exponential integrators, including the linearization process and the use of exact solutions to construct numerical schemes. It also covers the implementation of matrix function evaluations and provides an overview of applications in various fields such as quantum dynamics, chemistry, and mathematical finance. The authors conclude with a historical perspective on the development of exponential integrators.