The concept of “self-organized criticality” (SOC) was originally proposed as an explanation of 1/f-noise by Bak, Tang, and Wiesenfeld (1987), but turned out to have a far broader significance for scale-free nonlinear energy dissipation processes occurring in the entire universe. Over the last 30 years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into numerical SOC toy models. The novel applications stimulated also vigorous debates about the discrimination between SOC-related and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC models applied to astrophysical observations, attempt to describe what physics can be captured by SOC models, and offer a critique of weaknesses and strengths in existing SOC models.