The emergence of the first black holes during the first billion years of cosmic history marks a key event in cosmology. Their formation is part of the overall process of ending the cosmic dark ages, when the first stars appeared in low-mass dark matter haloes about a few 100 million years after the Big Bang. The first stars, galaxies, and black holes transformed the Universe from its simple initial state into one of ever increasing complexity. We review recent progress on our emerging theoretical picture of how the first black holes appeared on the cosmic scene, and how they impacted the subsequent history of the Universe. Our focus is on supermassive black holes, in particular assessing possible pathways to the formation of the billion-solar-mass black holes inferred to power the luminous quasars at high redshifts. We conclude with a discussion of upcoming empirical probes, such as the James Webb Space Telescope (JWST), and the Laser Interferometer Space Antenna (LISA), further ahead in time.

22 pages, 9 figures, Invited Review, Contemporary Physics, in press