Understanding the formation of the first stars is one of the frontier topics in modern astrophysics and cosmology .
Their emergence signalled the end of the cosmic dark ages , a few hundred million years after the Big Bang , leading to a fundamental transformation of the early Universe through the production of ionizing photons and the initial enrichment with heavy chemical elements .
We here review the state of our knowledge , separating the well understood elements of our emerging picture from those where more work is required .
Primordial star formation is unique in that its initial conditions can be directly inferred from the \Lambda Cold Dark Matter ( \Lambda CDM ) model of cosmological structure formation .
Combined with gas cooling that is mediated via molecular hydrogen , one can robustly identify the regions of primordial star formation , the so-called minihalos , having total masses of \sim 10 ^ { 6 } M _ { \odot } and collapsing at redshifts z \simeq 20 - 30 .
Within this framework , a number of studies have defined a preliminary standard model , with the main result that the first stars were predominantly massive .
This model has recently been modified to include a ubiquitous mode of fragmentation in the protostellar disks , such that the typical outcome of primordial star formation may be the formation of a binary or small multiple stellar system .
We will also discuss extensions to this standard picture due to the presence of dynamically significant magnetic fields , of heating from self-annihalating WIMP dark matter , or cosmic rays .
We conclude by discussing possible strategies to empirically test our theoretical models .
Foremost among them are predictions for the upcoming James Webb Space Telescope ( JWST ) , to be launched \sim 2018 , and for “ Stellar Archaeology ” , which probes the abundance pattern in the oldest , most-metal poor stars in our cosmic neighborhood , thereby constraining the nucleosynthesis inside the first supernovae .