Alternative polyadenylation (APA) is a major driver of transcriptome diversity in human cells. Here, we use deep learning to predict APA from DNA sequence alone. We trained our model (APARENT, APA REgression NeT) on isoform expression data from over 3 million APA reporters. APARENT's predictions are highly accurate when tasked with inferring APA in synthetic and human 3'UTRs. Visualizing features learned across all network layers reveals that APARENT recognizes sequence motifs known to recruit APA regulators, discovers previously unknown sequence determinants of 3' end processing, and integrates these features into a comprehensive, interpretable, cis-regulatory code. We apply APARENT to forward engineer functional polyadenylation signals with precisely defined cleavage position and isoform usage and validate predictions experimentally. Finally, we use APARENT to quantify the impact of genetic variants on APA. Our approach detects pathogenic variants in a wide range of disease contexts, expanding our understanding of the genetic origins of disease.