High energy density (HED) physics refers broadly to the study of macroscopic collections of matter under extreme conditions of temperature and density. The experimental facilities most widely used for these studies are high-power lasers and magnetic-pinch generators. The HED physics pursued at these facilities is still in its infancy, yet new regimes of experimental science are emerging. Examples from astrophysics include work relevant to planetary interiors, supernovae, astrophysical jets and accreting compact objects (such as neutron stars and black holes). In this paper, we review a selection of recent results in this new field of HED laboratory astrophysics and provide a brief look ahead to the coming decade.