Energy-dispersive photon detection with quantum microcalorimeters are reviewed emphasis with on the Transition Edge Sensor (TES) microcalorimeters. At X-ray wavelengths, these detectors realize high energy resolution and high quantum efficiency at the same time, and thus are very useful for observations of very weak cosmic X-rays. The signal response and the noise of the detectors are discussed. We show that they can detect photons of wide energy ranges; from infra-red to hard X-rays.