  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.
