These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248.