In this talk, we first recap the extended Dynamic Mode Decomposition (EDMD) as a very popular data-driven method to predict quantities of interest along the flow of a dynamical control system. To this end, the nonlinear dynamics are lifted into a high-, but finite-dimensional space, on which the surrogate model evolves linearly [1]. We embed EDMD in the Koopman framework to provide a rigorous error analysis depending on the amount of data by splitting up the approximation error into its two sources: projection and estimation [2]. Then, we provide a glimpse into a potential extension towards kernel EDMD (kEDMD). Here, we briefly touch upon the invariance of the respective reproducing kernel Hilbert space (RKHS) under the Koopman flow and the first uniform error bounds [3] for kEDMD. Finally, we present the usefulness of the EDMD-based surrogate model for (predictive) control and present novel results on closed-loop guarantees [3,4].