  From studying the cosmic microwave background, we know our Universe started
out very simple. It was by and large homogeneous and isotropic, with small
fluctuations that can be described by linear perturbation theory. In stark
contrast, the Universe today is highly structured on a vast range of length and
mass scales. In the evolution towards increasing complexity, the formation of
the first stars marks a primary transition event. The first generation of
stars, the so-called Population III (or Pop. III) build up from truly
metal-free primordial gas. They have long been thought to live short, solitary
lives, with only one massive star forming per halo. However, in recent years
this simple picture has undergone substantial revision, and we now understand
that stellar birth in the early Universe is subject to the same complexity as
star formation at present days. In this chapter, I review the current state of
the field. I begin by introducing the basics concepts of star-formation theory
and by discussing the typical environment in which Pop. III stars are thought
to form. Then I argue that the accretion disk that builds up in the center of a
halo is likely to fragment, resulting in the formation of a cluster of stars
with a wide range of masses, and I speculate about how this process may be
influenced by stellar feedback, the presence of magnetic fields, the energy
input from dark matter annihilation, and the occurrence of large- scale
streaming velocities between baryons and dark matter. Finally, I discuss direct
and indirect constraints on Pop. III star formation from high-redshift
observations and from the search for extremely metal-poor stars in the Milky
Way and its satellites.
