The importance of adaptive mutations in molecular evolution is extensively debated. Recent developments in population genomics allow inferring rates of adaptive mutations by fitting a distribution of fitness effects to the observed patterns of polymorphism and divergence at sites under selection and sites assumed to evolve neutrally. Here, we summarize the current state-of-the-art of these methods and review the factors that affect the molecular rate of adaptation. Several studies have reported extensive cross-species variation in the proportion of adaptive amino-acid substitutions (α) and predicted that species with larger effective population sizes undergo less genetic drift and higher rates of adaptation. Disentangling the rates of positive and negative selection, however, revealed that mutations with deleterious effects are the main driver of this population size effect and that adaptive substitution rates vary comparatively little across species. Conversely, rates of adaptive substitution have been documented to vary substantially within genomes. On a genome-wide scale, gene density, recombination and mutation rate were observed to play a role in shaping molecular rates of adaptation, as predicted under models of linked selection. At the gene level, it has been reported that the gene functional category and the macromolecular structure substantially impact the rate of adaptive mutations. Here, we deliver a comprehensive review of methods used to infer the molecular adaptive rate, the potential drivers of adaptive evolution and how positive selection shapes molecular evolution within genes, across genes within species and between species.