  This paper reviews Vlasov-based numerical methods used to model plasma in
space physics and astrophysics. Plasma consists of collectively behaving
charged particles that form the major part of baryonic matter in the Universe.
Many concepts ranging from our own planetary environment to the Solar system
and beyond can be understood in terms of kinetic plasma physics, represented by
the Vlasov equation. We introduce the physical basis for the Vlasov system, and
then outline the associated numerical methods that are typically used. A
particular application of the Vlasov system is Vlasiator, the world's first
global hybrid-Vlasov simulation for the Earth's magnetic domain, the
magnetosphere. We introduce the design strategies for Vlasiator and outline its
numerical concepts ranging from solvers to coupling schemes. We review
Vlasiator's parallelisation methods and introduce the used high-performance
computing (HPC) techniques. A short review of verification, validation and
physical results is included. The purpose of the paper is to present the Vlasov
system and introduce an example implementation, and to illustrate that even
with massive computational challenges, an accurate description of physics can
be rewarding in itself and significantly advance our understanding. Upcoming
supercomputing resources are making similar efforts feasible in other fields as
well, making our design options relevant for others facing similar challenges.
