The formation of the first stars in the universe

In this review, I survey our current understanding of how the very first stars in the universe formed, with a focus on three main areas of interest: the formation of the first protogalaxies and the cooling of gas within them, the nature and extent of fragmentation within the cool gas, and the physics - in particular the interplay between protostellar accretion and protostellar feedback - that serves to determine the final stellar mass. In each of these areas, I have attempted to show how our thinking has developed over recent years, aided in large part by the increasing ease with which we can now perform detailed numerical simulations of primordial star formation. I have also tried to indicate the areas where our understanding remains incomplete, and to identify some of the most important unsolved problems.