Open questions in the study of population III star formation

The first stars were key drivers of early cosmic evolution. We review the main physical elements of the current consensus view, positing that the first stars were predominantly very massive. We continue with a discussion of important open questions that confront the standard model. Among them are uncertainties in the atomic and molecular physics of the hydrogen and helium gas, the multiplicity of stars that form in minihalos, and the possible existence of two separate modes of metal-free star formation.