Abstract: Reaction-diffusion waves describe diverse natural phenomena from crystal growth in physics to tumor growth in biology. Many aspects of these phenomena are stochastic because expanding entities---whether atoms or cells---are discrete. A quantitative description of fluctuations in growing populations remains an open problem. Recently, we have made significant progress in this direction by looking at physical processes through the lens of biology. I will show that expanding populations fall into one of three universality classes with very different physical and genetic properties. For example, genealogical trees have different structure, and fluctuations scale differently with the population density. Surprisingly, scaling exponents and many other properties can be computed exactly. On the biology side, our theory predicts that positive density-dependence in growth or dispersal could dramatically alter evolution in expanding populations even when its contribution to the expansion velocity is small. On the mathematics side, our work highlights potential pitfalls in the commonly-used method to approximate stochastic dynamics and shows how to avoid them.