The big problems in star formation: The star formation rate, stellar clustering, and the initial mass function

Star formation lies at the center of a web of processes that drive cosmic evolution: generation of radiant energy, synthesis of elements, formation of planets, and development of life. Decades of observations have yielded a variety of empirical rules about how it operates, but at present we have no comprehensive, quantitative theory. In this review I discuss the current state of the field of star formation, focusing on three central questions: What controls the rate at which gas in a galaxy converts to stars? What determines how those stars are clustered, and what fraction of the stellar population ends up in gravitationally-bound structures? What determines the stellar initial mass function, and does it vary with star-forming environment? I use these three questions as a lens to introduce the basics of star formation, beginning with a review of the observational phenomenology and the basic physical processes. I then review the status of current theories that attempt to solve each of the three problems, pointing out links between them and opportunities for theoretical and numerical work that crosses the scale between them. I conclude with a discussion of prospects for theoretical progress in the coming years.