  If Dark Matter (DM) is composed by Weakly Interacting Massive Particles, its
annihilation in the halos harboring the earliest star formation episode may
strongly influence the first generation of stars (Population III). Whereas DM
annihilation at early stages of gas collapse does not dramatically affect the
properties of the cloud, the formation of a hydrostatic object (protostar) and
its evolution toward the main sequence may be delayed. This process involves DM
concentrated in the center of the halo by gravitational drag, and no consensus
is yet reached over whether this can push the initial mass of Population III to
higher masses. DM can also be captured through scattering over the baryons in a
dense object, onto or very close to the Main Sequence. This mechanism can
affect formed stars and in principle prolonge their lifetimes. The strength of
both mechanisms depends upon several environmental conditions and on DM
parameters; such spread in the parameter space leads to very different
scenarios for the observables in the Population. Here I summarize the state of
the art in modelling and observational expectations, eventually highlighting
the most critical assumptions and sources of uncertainty.
