ABSTRACT:  In this tutorial, we will discuss the theory and computation of Koopman operators. The Koopman operator is a linear, infinite-dimensional representation of a nonlinear dynamical system. This linear representation allows for a spectral analysis of a nonlinear system. We will give an overview of the basic properties, and the state-of-the-art methods for the computation of the Koopman operator, such as dynamic mode decomposition (DMD) and its variants. In the second half of the talk, we will discuss some open problems in relation to the computation of the Koopman operator in high-dimensional systems like numerical simulations of fluid flow. In particular, we will explore the difficulty of choosing an appropriate basis for extended DMD and possible connections between the Koopman operator spectrum and the ergodic properties of the dynamics.