100 years ago, Karl Schwarzschild solved Einstein’s equations for general relativity and obtained a solution that described a black hole five decades before the first astronomical evidence that they exist. As exotic structures of spacetime, black holes continue to fascinate astronomers, physicists and the general public. I will summarize the fundamental role that black holes play in astrophysics, producing the most luminous objects in the universe—seen billions of light-years from Earth—and ruling over the fates of galaxies. I will then list the big open questions in the field and describe recent progress, particularly in numerical simulations of black hole accretion