Simulations of cosmic structure formation address multi-scale, multi-physics problems of vast proportions. These calculations are presently at the forefront of today's use of supercomputers, and are important scientific drivers for the future use of exaflop computing platforms. However, continued future success in this field requires the development of new numerical methods that excel in accuracy, robustness, parallel scalability, and physical fidelity to the processes relevant in galaxy and star formation. In an interdisciplinary and international effort of astrophysicists and applied mathematicians we will in this project substantially improve the astrophysical moving-mesh code AREPO and extend its range of applicability, with the goal of producing an internationally leading application code for the upcoming large computing platforms. We work on new, powerful high-order discontinuous Galerkin schemes, on more efficient solvers for gravity and for anisotropic transport of heat and relativistic particles, and on an improvement of the accuracy of the treatment of ideal magnetohydrodynamics. We aim to drastically enhance the raw performance and scalability of the code by employing sophisticated hybrid parallelisation techniques combined with low-level optimizations that make full use of vector instructions and device accelerators. We will apply our code on current state-of-the art supercomputers to carry out transformative magnetohydrodynamic simulations of galaxy and primordial star formation, stretching the envelope of what is possible today and in the years to come. We will also work towards publicly releasing the AREPO and GADGET-4 codes.