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Automated sleep stage classification using heart-rate variability is an active field of research. In this work limitations of the current state-of-the-art are addressed through the use of deep learning techniques and their efficacy is demonstrated. First, a temporal model is proposed for the inference of sleep stages from electrocardiography using a deep long- and short-term (LSTM) classifier and it is shown that this model outperforms previous approaches which were often limited to non-temporal or Markovian classifiers on a comprehensive benchmark data set (292 participants, 541214 samples) comprising a wide range of ages and pathological profiles, achieving a Cohen's $\kappa$ of $0.61\pm0.16$ and accuracy of $76.30\pm10.17$ annotated according to the Rechtschaffen & Kales annotation standard. Subsequently, it is demonstrated how knowledge learned on this large benchmark data set can be re-used through transfer learning for the classification of photoplethysmography (PPG) data. This is done using a smaller data set (60 participants, 91479 samples) that is annotated with the more recent American Association of Sleep Medicine annotation standard, achieving a Cohen's $\kappa$ of $0.63\pm0.13$ and accuracy of $74.65\pm8.63$ for wrist-mounted PPG-based sleep stage classification, higher than any previously reported performance using this sensor modality. This demonstrates the feasibility of knowledge transfer in sleep staging to adapt models for new sensor modalities as well as different annotation strategies.