Quantifying the pathogenicity of protein variants in human disease-related genes would have a marked effect on clinical decisions, yet the overwhelming majority (over 98%) of these variants still have unknown consequences. In principle, computational methods could support the large-scale interpretation of genetic variants. However, state-of-the-art methods have relied on training machine learning models on known disease labels. As these labels are sparse, biased and of variable quality, the resulting models have been considered insufficiently reliable. Here we propose an approach that leverages deep generative models to predict variant pathogenicity without relying on labels. By modelling the distribution of sequence variation across organisms, we implicitly capture constraints on the protein sequences that maintain fitness. Our model EVE (evolutionary model of variant effect) not only outperforms computational approaches that rely on labelled data but also performs on par with, if not better than, predictions from high-throughput experiments, which are increasingly used as evidence for variant classification. We predict the pathogenicity of more than 36 million variants across 3,219 disease genes and provide evidence for the classification of more than 256,000 variants of unknown significance. Our work suggests that models of evolutionary information can provide valuable independent evidence for variant interpretation that will be widely useful in research and clinical settings.