High performance computing numerical simulations are today one of the more effective instruments to implement and study new theoretical models, and they are mandatory during the preparatory phase and operational phase of any scientific experiment. New challenges in Cosmology and Astrophysics will require a large number of new extremely computationally intensive simulations to investigate physical processes at different scales. Moreover, the size and complexity of the new generation of observational facilities also implies a new generation of high performance data reduction and analysis tools pushing toward the use of Exascale computing capabilities. Exascale supercomputers cannot be produced today. We discuss the major technological challenges in the design, development and use of such computing capabilities and we will report on the progresses that has been made in the last years in Europe, in particular in the framework of the ExaNeSt European funded project. We also discuss the impact of this new computing resources on the numerical codes in Astronomy and Astrophysics.