The study of star formation is currently benefiting from a wealth of new observational data, exploiting the high-sensitivity, wide-field, high-resolution capabilities of a diverse range of space and ground-based instrumentation. In parallel with this, high performance computing is enabling theorists to tackle key problems which—due to their complex geometry and non-linear nature—had long been recognized to be beyond the reach of analytical theory. In this review, rather than reporting progress in each of these areas, I will instead set out some scientific questions that one would expect to be answered before one would regard star formation as a topic that was largely solved. I have accordingly selected three areas: 1) molecular clouds and their relationship to the stars they form and to the wider galactic disk, 2) the question of the determinants of stellar mass (i.e., the IMF), and 3) the issue of protostellar disk dispersal and its relation to planet formation). For each topic, I outline areas of consensus, recent results, and discuss the key problems that can plausibly be addressed in the next five years.