Environmental changes have caused episodes of habitat expansions in the evolutionary history of many species. These range changes affect the dynamics of biological evolution in multiple ways. Recent microbial experiments as well as simulations suggest that enhanced genetic drift at the frontier of a two-dimensional range expansion can cause genetic sectoring patterns with fractal domain boundaries. Here, we propose and analyze a simple model of asexual biological evolution at expanding frontiers that explains these neutral patterns and predicts the effect of natural selection. We find that beneficial mutations give rise to sectors with an opening angle that depends sensitively on the selective advantage of the mutants. Deleterious mutations, on the other hand, are not able to establish a sector permanently. They can, however, temporarily "surf" on the population front, and thereby reach unusually high frequencies. As a consequence, expanding frontiers are loaded with a high fraction of mutants at mutation-selection balance. Numerically, we also determine the condition at which the wild type is lost in favor of deleterious mutants (genetic meltdown) at a growing front. Our prediction for this error threshold differs qualitatively from existing well-mixed theories, and sets tight constraints on sustainable mutation rates for populations that undergo frequent range expansions.