A high-energy e + e − Linear Collider has been considered since a long time as an important complement to the LHC. Unprecedented precision measurements as well as the exploration of so far untouched phase space for direct production of new particles will provide unique information to advance the limits of our understanding of our universe. Within this project, the physics prospects of such a collider as well as their interplay with design of the accelerator and the detectors have been investigated in a quantitative way. This kind of study required a close collaboration between theory and experiment, always taking into account results of the LHC and other relevant experiments. In this article we will summarize some of the most important developments and results, covering all core areas of the physics progamme of future e + e − colliders.