In this work we introduce NeoCam, an open source hardware-software platform for video-based monitoring of preterms infants in Neonatal Intensive Care Units (NICUs). NeoCam includes an edge computing device that performs video acquisition and processing in real-time. Compared to other proposed solutions, it has the advantage of handling data more efficiently by performing most of the processing on the device, including proper anonymisation for better compliance with privacy regulations. In addition, it allows to perform various video analysis tasks of clinical interest in parallel at speeds of between 20 and 30 frames-per-second. We introduce algorithms to measure without contact the breathing rate, motor activity, body pose and emotional status of the infants. For breathing rate, our system shows good agreement with existing methods provided there is sufficient light and proper imaging conditions. Models for motor activity and stress detection are new to the best of our knowledge. NeoCam has been tested on preterms in the NICU of the University Hospital Puerta del Mar (Cádiz, Spain), and we report the lessons learned from this trial.