Summary form only given. Large-scale computation has achieved a pivotal role in many areas of astrophysics and cosmology. Not only does increasing computational power allow us to better model the huge range of spatial and temporal scales in the universe, but simulations provide in many cases the only laboratory in which we may actively experiment on cosmic phenomena. The talk will highlight some of the significant achievements in numerical astrophysics and cosmology over the past decade and describe some of the key challenges that we must face in the future. It will focus, in particular, on some recent state-of-the-art models of dwarf galaxies and illustrate the subtle interplay between gravitational and astrophysical processes in these systems.