Magnetic-fusion plasmas are complex self-organized systems with an extremely wide range of spatial and temporal scales, from the electron-orbit scales (⇠10 11 s, ⇠10 5 m) to the di usion time of electrical current through the plasma (⇠10 2 s) and the distance along the magnetic field between two solid surfaces in the region that determines the plasma-wall interactions (⇠100 m). The description of the individual phenomena and of the nonlinear coupling between them involves a hierarchy of models, which, when applied to realistic configurations, require the most advanced numerical techniques and algorithms and the use of state-of-the-art high-performance computers. The common thread of such models resides in the fact that the plasma components are at the same time sources of electromagnetic fields, via the charge and current densities that they generate, and subject to the action of electromagnetic fields. This leads to a wide variety of plasma modes of oscillations that resonate with the particle or fluid motion and makes the plasma dynamics much richer than that of conventional, neutral fluids.T he most straightforward way for describing plasmas would be the microscopic particle approach: solving the equations of motion for the many individual particles that form the plasma in externally imposed electromagnetic fields and in the fields that the particles themselves generate. However, this is computationally impossible to apply to realistic magnetic-fusion plasmas, which typically contain 10 22 -10 23 particles. (For a review of the basic concepts of magnetically confined fusion plasmas, see ref. 1.)A statistical treatment leads to the kinetic model, based on the Vlasov equation (equation (1)), essentially Liouville's theorem applied to the specific plasma environment, with interactions that are dominated by electromagnetic fields and a separation between particle collisions -that is, interactions at microscopic scales-and long-range fields. The Vlasov equation describes the evolution of f s (x,v,t), the single-particle phase-space distribution function of the species labelled 's' (electrons or ions), under the e ect of the longrange electric and magnetic fields E and B, which evolve according to Maxwell's equations with plasma charge and current densities as sources:Here, x, v, q s and m s stand for the position, velocity, charge and the mass of a particle of species 's' , respectively. In the Vlasov model, collisions are neglected, an approximation that is generally justified for fusion plasmas because small-scale e ects based on Coulomb interactions become very weak at high temperatures. In fact, Coulomb collisions' cross-sections for the exchange of momentum and energy, and related quantities such as the electrical resistivity ⌘, scale with T 3/2 . In ITER core plasmas (see ref. 1), for example, T ⇡ 10 keV (⇡10 8 K) and the electron-electron ('ee'), electron-ion ('ei') and ion-ion ('ii') collisional exchanges of momentum and energy ('E') occur over relatively long characteristic times:2)-justifying the collisi...