Teasing apart the effects of selection and demography on genetic polymorphism remains one of the major challenges in the analysis of population genomic data. The traditional approach has been to assume that demography would leave a genome-wide signature, whereas the effect of selection would be local. In the light of recent genomic surveys of sequence polymorphism, several authors have argued that this approach is questionable based on the evidence of the pervasive role of positive selection and that new approaches are needed. In the first part of this review, we give a few empirical and theoretical examples illustrating the difficulty in teasing apart the effects of selection and demography on genomic polymorphism patterns. In the second part, we review recent efforts to detect recent positive selection. Most available methods still rely on an a priori classification of sites in the genome but there are many promising new approaches. These new methods make use of the latest developments in statistics, explore aspects of the data that had been neglected hitherto or take advantage of the emerging population genomic data. A current and promising approach is based on first estimating demographic and genetic parameters, using, e.g., a likelihood or approximate Bayesian computation framework, focusing on extreme outlier regions, and then using an independent method to confirm these. Finally, especially for species where evidence of natural selection has been limited, more experimental and versatile approaches that contrast populations under varied environmental constraints might be more successful compared with species-wide genome scans in search of specific signatures.