  The first stars to form in the Universe -- the so-called Population III stars
-- bring an end to the cosmological Dark Ages, and exert an important influence
on the formation of subsequent generations of stars and on the assembly of the
first galaxies. Developing an understanding of how and when the first
Population III stars formed and what their properties were is an important goal
of modern astrophysical research. In this review, I discuss our current
understanding of the physical processes involved in the formation of Population
III stars. I show how we can identify the mass scale of the first dark matter
halos to host Population III star formation, and discuss how gas undergoes
gravitational collapse within these halos, eventually reaching protostellar
densities. I highlight some of the most important physical processes occurring
during this collapse, and indicate the areas where our current understanding
remains incomplete. Finally, I discuss in some detail the behaviour of the gas
after the formation of the first Population III protostar. I discuss both the
conventional picture, where the gas does not undergo further fragmentation and
the final stellar mass is set by the interplay between protostellar accretion
and protostellar feedback, and also the recently advanced picture in which the
gas does fragment and where dynamical interactions between fragments have an
important influence on the final distribution of stellar masses.
