Single-photon imaging detectors promise the ultimate in sensitivity by eliminating read noise. These devices could provide extraordinary benefits for photon-starved applications, e.g., imaging exoplanets, fast wavefront sensing, and probing the human body through transluminescence. Recent implementations are often in the form of sparse arrays that have less-than-unity fill factor. For imaging, fill factor is typically enhanced by using microlenses, at the expense of photometric and spatial information loss near the edges and corners of the pixels. Other challenges include afterpulsing and the potential for photon self-retriggering. Both effects produce spurious signal that can degrade the signal-to-noise ratio. This paper reviews development and potential application of single-photon-counting detectors, including highlights of initiatives in the Center for Detectors at the Rochester Institute of Technology and MIT Lincoln Laboratory. Current projects include single-photon-counting imaging detectors for the Thirty Meter Telescope, a future NASA terrestrial exoplanet mission, and imaging LIDAR detectors for planetary and Earth science space missions.