Koopman operators provide a linear operator approach to nonlinear dynamics via their spectral properties. Extracting this information from trajectory data has become a significant focus, yet standard discretisation-based methods often fail. Chapter 11 builds a rigorous, practitioner-ready framework for data-driven Koopman spectral analysis, with case studies spanning chaotic and Hamiltonian dynamics, climate systems, shockwaves, and turbulent flows. After a focused primer, we analyse DMD and EDMD and present concrete failures on benchmark systems such as the Lorenz and Duffing systems. We then introduce Residual DMD (ResDMD), a data-driven analogue of operator folding that uses residuals to certify eigenpairs and reject spurious eigenvalues; kernelised variants extend the approach to high-dimensional settings via RKHS observables and a dual viewpoint through the adjoint Koopman operator. For measure-preserving dynamics, we develop mpEDMD and smoothing schemes based on correlations and resolvents to compute spectral measures and spectral type. Finally, adversarial examples and SCI lower bounds expose intrinsic limits of Koopman learning, including for random trajectory sampling.