Our Universe is filled with plasma which often manifests itself via various spectacular phenomena. Our understanding of many of them has benefitted tremendously over the past decades from direct application of classic plasma physics results originally obtained in traditional plasma physics areas like space physics and magnetic fusion. Often, however, the classical plasma physics falls short, becomes inapplicable in many high-energy astrophysics situations. The physical conditions in plasmas surrounding relativistic compact objects like black holes and neutron stars are quite extreme, and the effects of special and general relativity, radiation, pair production, ultra-strong magnetic fields — effects viewed as exotic by a traditional plasma physicist — become critical. Understanding how various plasma processes (waves and instabilities, magnetic reconnection, turbulence, shocks) operate under such extreme conditions is thus an important new research frontier. I will present the current status of this emerging and exciting branch of plasma astrophysics and will review the rapid progress it has enjoyed in recent years. This progress is made possible by a combination of concerted theoretical efforts and recent computational breakthroughs, including the advent of plasma codes that now self-consistently incorporate new physics capabilities like synchrotron and inverse-Compton radiation and its back-reaction, QED pair-creation processes, and general relativity. I will give several examples of how recent first-principles numerical studies of plasma processes under extreme conditions have helped resolve some puzzles in high-energy astrophysics, and will also outline promising future directions of research.