Population waves have played a crucial role in evolutionary history, as in the 'out of Africa' hypothesis for human ancestry. Population geneticists and physicists are now developing methods for understanding how mutations, number fluctuations and selective advantages play out in such situations. Once the behavior of pioneer organisms at frontiers is understood, genetic markers can be used to infer information about growth, ancestral population size and colonization pathways. Insights into the nature of competition and cooperation at frontiers are possible. Neutral mutations optimally positioned at the front of a growing population wave can increase their abundance by 'surfing' on the population wave. In addition, obstacles such as lakes, deserts and mountains alter migration fronts and organism geneologies in important and interesting ways, which can be illuminated by a kind of 'Huygens Principle' for biological waves. Experimental and theoretical studies of these effects will be presented, using bacteria and yeast as model systems.