We review recent theoretical results on the formation of the first stars in
the universe, and emphasize related open questions. In particular, we discuss
the initial conditions for Population III star formation, as given by variants
of the cold dark matter cosmology. Numerical simulations have investigated the
collapse and the fragmentation of metal-free gas, showing that the first stars
were predominantly very massive. The exact determination of the stellar masses,
and the precise form of the primordial initial mass function, is still hampered
by our limited understanding of the accretion physics and the protostellar
feedback effects. We address the importance of heavy elements in bringing about
the transition from an early star formation mode dominated by massive stars, to
the familiar mode dominated by low mass stars, at later times. We show how
complementary observations, both at high redshifts and in our local cosmic
neighborhood, can be utilized to probe the first epoch of star formation.Comment: 38 pages, 10 figures, draft version for 2004 Annual Reviews of
Astronomy and Astrophysics, high-resolution version available at
http://cfa-www.harvard.edu/~vbromm