Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra, can lack finite-dimensional invariant subspaces, and approximations can suffer from spectral pollution (spurious modes). These issues make computing the spectral properties of Koopman operators a considerable challenge. This two- part talk will detail the first scheme (ResDMD) with convergence guarantees for computing the spectra and pseudospectra of general Koopman operators from snapshot data. Furthermore, we use the resolvent operator and ResDMD to compute smoothed approximations of spectral measures (including continuous spectra), with explicit high-order convergence. ResDMD is similar to extended DMD, except it rigorously concurrently computes a residual from the same snapshot data, allowing practitioners to gain confidence in the computed results. Kernelized variants of our algorithms allow for dynamical systems with a high- dimensional state-space, and the error control provided by ResDMD allows a posteriori verification of learnt dictionaries. We apply ResDMD to compute the spectral measure associated with the dynamics of a protein molecule (20,046-dimensional state-space) and demonstrate several problems in fluid dynamics (with state-space dimensions > 100,000). For example, we compare ResDMD and DMD for particle image velocimetry data from turbulent wall-jet flow, the acoustic signature of laser-induced plasma, and turbulent flow past a cascade of aerofoils.