Breast cancer remains a global challenge, causing over 600,000 deaths in 2018 (ref. (1)). To achieve earlier cancer detection, health organizations worldwide recommend screening mammography, which is estimated to decrease breast cancer mortality by 20-40% (refs. (2,3)). Despite the clear value of screening mammography, significant false positive and false negative rates along with non-uniformities in expert reader availability leave opportunities for improving quality and access(4,5). To address these limitations, there has been much recent interest in applying deep learning to mammography(6-18), and these efforts have highlighted two key difficulties: obtaining large amounts of annotated training data and ensuring generalization across populations, acquisition equipment and modalities. Here we present an annotation-efficient deep learning approach that (1) achieves state-of-the-art performance in mammogram classification, (2) successfully extends to digital breast tomosynthesis (DBT; '3D mammography'), (3) detects cancers in clinically negative prior mammograms of patients with cancer, (4) generalizes well to a population with low screening rates and (5) outperforms five out of five full-time breast-imaging specialists with an average increase in sensitivity of 14%. By creating new 'maximum suspicion projection' (MSP) images from DBT data, our progressively trained, multiple-instance learning approach effectively trains on DBT exams using only breast-level labels while maintaining localization-based interpretability. Altogether, our results demonstrate promise towards software that can improve the accuracy of and access to screening mammography worldwide.