Plasma turbulence is a ubiquitous phenomenon, influencing the dynamics in most of the visible universe and playing a crucial role in countless laboratory experiments of basic or applied plasma science. Yet, various fundamental aspects of this prototypical nonlinear process involving many degrees of freedom and leading to self-organization far from thermodynamic equilibrium are only poorly understood at present. The aim of this project is to tackle a number of longstanding unsolved problems related to plasma turbulence by means of extreme computing approaching the exascale. For this purpose, a novel generation of numerical tools will be developed and run on some of the largest supercomputers in the years to come, breaking new ground both scientifically as well as computationally. The three main goals are to develop simulation capabilities for peta- to exascale computations of plasma turbulence, to strive to unravel the general nature of plasma turbulence, and to better grasp the creation of magnetic fields in turbulent plasmas together with their effects on energetic particles. Exploiting the fact that the same basic processes are operative in laboratory plasmas as well as throughout the plasma universe, and involving a critical mass of scientists with a strong background in plasma physics, astrophysics, and applied mathematics, the project is well set up for frontier research based on fruitful interactions between these neighbouring fields. Most importantly, it will lead to a new level of understanding of turbulence in laboratory and astrophysical plasmas and help pioneer the use of the next generation(s) of supercomputers – both of which will be of wide benefit.