Study Objective: Sleep is reflected not only in the electroencephalogram but also in heart rhythms and breathing patterns. Therefore, we hypothesize that it is possible to accurately stage sleep based on the electrocardiogram (ECG) and respiratory signals. Methods: Using a dataset including 8,682 polysomnographs, we develop deep neural networks to stage sleep from ECG and respiratory signals. Five deep neural networks consisting of convolutional networks and long short-term memory networks are trained to stage sleep using heart and breathing, including the timing of R peaks from ECG, abdominal and chest respiratory effort, and the combinations of these signals. Results: ECG in combination with the abdominal respiratory effort achieve the best performance for staging all five sleep stages with a Cohen's kappa of 0.600 (95% confidence interval 0.599 -- 0.602); and 0.762 (0.760 -- 0.763) for discriminating awake vs. rapid eye movement vs. non-rapid eye movement sleep. The performance is better for young participants and for those with a low apnea-hypopnea index, while it is robust for commonly used outpatient medications. Conclusions: Our results validate that ECG and respiratory effort provide substantial information about sleep stages in a large population. It opens new possibilities in sleep research and applications where electroencephalography is not readily available or may be infeasible, such as in critically ill patients.