NOTICE: Special Colloquium Date
Massive stars, those larger than 20 times the mass of the Sun, are very rare, but their extreme luminosities make them both the only type of young star we can observe in distant galaxies and the dominant energy sources in the universe today. How such stars form, however, is a longstanding mystery. Efficient radiative cooling in the dense clouds where massive stars form should favor fragmentation into objects of solar mass or smaller. Even if a collapsing cloud exceeds this mass, massive stars produce so much light that the radiation pressure they exert on the gas and dust around them is stronger than their gravitational attraction, a condition that has long been expected to prevent them from growing by accretion. Unraveling these mysteries requires new techniques for simulating gravito-radiation-hydrodynamics in three-dimensions over large dynamic ranges. I introduce these methods, and I present results from recent simulations that point toward solutions to these problems.