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 question 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.
