Heart rate (HR) is one of the essential vital signs used to indicate the physiological health of the human body. While traditional HR monitors usually require contact with skin, remote photoplethysmography (rPPG) enables contactless HR monitoring by capturing subtle light changes of skin through a video camera. Given the vast potential of this technology in the future of digital healthcare, remote monitoring of physiological signals has gained significant traction in the research community. In recent years, the success of deep learning (DL) methods for image and video analysis has inspired researchers to apply such techniques to various parts of the remote physiological signal extraction pipeline. In this paper, we discuss several recent advances of DL-based methods specifically for remote HR measurement, categorizing them based on model architecture and application. We further detail relevant real-world applications of remote physiological monitoring and summarize various common resources used to accelerate related research progress. Lastly, we analyze the implications of research findings and discuss research gaps to guide future explorations.