The nearly neutral theory represents a development of Kimura's neutral theory of molecular evolution that makes testable predictions that go beyond a mere null model. Recent evidence has strongly supported several of these predictions, including the prediction that slightly deleterious variants will accumulate in a species that has undergone a severe bottleneck or in cases where recombination is reduced or absent. Because bottlenecks often occur in speciation and slightly deleterious mutations in coding regions will usually be nonsynonymous, we should expect that the ratio of nonsynonymous to synonymous fixed differences between species should often exceed the ratio of nonsynonymous to synonymous polymorphisms within species. Many data support this prediction, although they have often been wrongly interpreted as evidence for positive Darwinian selection. The use of conceptually flawed tests for positive selection has become widespread in recent years, seriously harming the quest for an understanding of genome evolution. When properly analyzed, many (probably most) claimed cases of positive selection will turn out to involve the fixation of slightly deleterious mutations by genetic drift in bottlenecked populations. Slightly deleterious variants are a transient feature of evolution in the long term, but they have substantially affected contemporary species, including our own.