Experimental evolutionary genomics now allows biologists to test fundamental theories concerning the genetic basis of adaptation. We have conducted one of the longest laboratory evolution experiments with any sexually-reproducing metazoan, Drosophila melanogaster. We used next-generation resequencing data from this experiment to examine genome-wide patterns of genetic variation over an evolutionary time-scale that approaches 1,000 generations. We also compared measures of variation within and differentiation between our populations to simulations based on a variety of evolutionary scenarios. Our analysis yielded no clear evidence of hard selective sweeps, whereby natural selection acts to increase the frequency of a newly-arising mutation in a population until it becomes fixed. We do find evidence for selection acting on standing genetic variation, as independent replicate populations exhibit similar population-genetic dynamics, without obvious fixation of candidate alleles under selection. A hidden-Markov model test for selection also found widespread evidence for selection. We found more genetic variation genome-wide, and less differentiation between replicate populations genome-wide, than arose in any of our simulated evolutionary scenarios.