The HyTeG finite-element software framework for scalable multigrid solvers. In this article, a new generic higher-order finite-element framework for massively parallel simulations is presented. The modular software architecture is carefully designed to exploit the resources of modern and future supercomputers. Combining an unstructured topology with structured grid refinement facilitates high geometric adaptability and matrix-free multigrid implementations with excellent performance. Different abstraction levels and fully distributed data structures additionally ensure high flexibility, extensibility, and scalability. The software concepts support sophisticated load balancing and flexibly combining finite-element spaces. Example scenarios with coupled systems of partial differential equations show the applicability of the concepts to performing geophysical simulations.