Copy number variants (CNVs), deletions and duplications of segments of DNA, account for at least five times more variable base pairs in humans than single-nucleotide variants. Several common CNVs were shown to change coding and regulatory sequences and thus dramatically affect adaptive phenotypes involving immunity, perception, metabolism, skin structure, among others. Some of these CNVs were also associated with susceptibility to cancer, infection, and metabolic disorders. These observations raise the possibility that CNVs are a primary contributor to human phenotypic variation and consequently evolve under selective pressures. Indeed, locus-specific haplotype-level analyses revealed signatures of natural selection on several CNVs. However, more traditional tests of selection which are often applied to single-nucleotide variation often have diminished statistical power when applied to CNVs because they often do not show strong linkage disequilibrium with nearby variants. Recombination-based formation mechanisms of CNVs lead to frequent recurrence and gene conversion events, breaking the linkage disequilibrium involving CNVs. Similar methodological challenges also prevent routine genome-wide association studies to adequately investigate the impact of CNVs on heritable human disease. Thus, we argue that the full relevance of CNVs to human health and evolution is yet to be elucidated. We further argue that a holistic investigation of formation mechanisms within an evolutionary framework would provide a powerful framework to understand the functional and biomedical impact of CNVs. In this paper, we review several cases where studies reveal diverse evolutionary histories and unexpected functional consequences of CNVs. We hope that this review will encourage further work on CNVs by both evolutionary and medical geneticists.