This paper presents a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator, which governs the evolution of scalar observables in autonomous dynamical systems. The method, called Extended Dynamic Mode Decomposition (EDMD), requires a dataset of snapshot pairs and a dictionary of scalar observables, without needing explicit governing equations or interaction with a "black box" integrator. EDMD extends Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. If the data are generated by a Markov process, EDMD approximates the eigenfunctions of the Kolmogorov backward equation, known as the "stochastic Koopman operator." The paper demonstrates the effectiveness of EDMD on both deterministic and stochastic examples, showing its ability to approximate Koopman eigenfunctions and modes even with limited data. EDMD is shown to converge to a Galerkin method in the limit of large data, and its performance is validated on various examples, including a linear system and the unforced Duffing oscillator. The method is also shown to be applicable to stochastic systems, where it approximates the eigenfunctions of the stochastic Koopman operator. The paper concludes that EDMD provides a powerful tool for analyzing and decomposing nonlinear dynamical systems, with potential applications in model reduction and nonlinear manifold learning.This paper presents a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator, which governs the evolution of scalar observables in autonomous dynamical systems. The method, called Extended Dynamic Mode Decomposition (EDMD), requires a dataset of snapshot pairs and a dictionary of scalar observables, without needing explicit governing equations or interaction with a "black box" integrator. EDMD extends Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. If the data are generated by a Markov process, EDMD approximates the eigenfunctions of the Kolmogorov backward equation, known as the "stochastic Koopman operator." The paper demonstrates the effectiveness of EDMD on both deterministic and stochastic examples, showing its ability to approximate Koopman eigenfunctions and modes even with limited data. EDMD is shown to converge to a Galerkin method in the limit of large data, and its performance is validated on various examples, including a linear system and the unforced Duffing oscillator. The method is also shown to be applicable to stochastic systems, where it approximates the eigenfunctions of the stochastic Koopman operator. The paper concludes that EDMD provides a powerful tool for analyzing and decomposing nonlinear dynamical systems, with potential applications in model reduction and nonlinear manifold learning.