The paper presents a data-driven method for approximating the Koopman operator, which governs the evolution of scalar observables in nonlinear dynamical systems. The method, referred to as Extended Dynamic Mode Decomposition (EDMD), uses a dataset of snapshot pairs and a dictionary of scalar observables to approximate the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. EDMD is shown to be an extension of Dynamic Mode Decomposition (DMD) and can produce better approximations of the Koopman eigenfunctions. The paper also demonstrates that EDMD converges to a Galerkin method in the limit of large data and provides examples of its effectiveness on deterministic and stochastic data. Additionally, the paper discusses the choice of dictionary and its impact on the accuracy of the approximations.The paper presents a data-driven method for approximating the Koopman operator, which governs the evolution of scalar observables in nonlinear dynamical systems. The method, referred to as Extended Dynamic Mode Decomposition (EDMD), uses a dataset of snapshot pairs and a dictionary of scalar observables to approximate the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. EDMD is shown to be an extension of Dynamic Mode Decomposition (DMD) and can produce better approximations of the Koopman eigenfunctions. The paper also demonstrates that EDMD converges to a Galerkin method in the limit of large data and provides examples of its effectiveness on deterministic and stochastic data. Additionally, the paper discusses the choice of dictionary and its impact on the accuracy of the approximations.