Senescence has long been a public health challenge as well as a fascinating evolutionary problem. There is neither a universally accepted theory for its ultimate causes, nor a consensus about what may be its impact on human health. Here we test the predictions of two evolutionary explanations of senescence - mutation accumulation and antagonistic pleiotropy - which postulate that genetic variants with harmful effects in old ages can be tolerated, or even favoured, by natural selection at early ages. Using data from genome-wide association studies (GWAS), we study the effects of genetic variants associated with diseases appearing at different periods in life, when they are expected to have different impacts on fitness. Data fit theoretical expectations. Namely, we observe higher risk allele frequencies combined with large effect sizes for late-onset diseases, and detect a significant excess of early-late antagonistically pleiotropic variants that, strikingly, tend to be harboured by genes related to ageing. Beyond providing systematic, genome-wide evidence for evolutionary theories of senescence in our species and contributing to the long-standing question of whether senescence is the result of adaptation, our approach reveals relationships between previously unrelated pathologies, potentially contributing to tackling the problem of an ageing population.