This volume describes the physics behind the building block of a new type of X-ray camera. This camera will be able to take pictures of astronomical objects that emit X-rays, and at the same time measure the spectrum for each pixel in the image. This is accomplished by means of a device called a superconducting transition-edge microcalorimeter. This tiny device can measure the energy of a single X-ray photon with an accuracy better than a tenth of a percent. This book describes the design, manufacture and read-out of such a device. In addition, a theoretical model of the device is presented which can be used for optimisation of the design parameters. In the appendix, an overview is given of digital filtering techniques for single- photon pulse data. The book is a complete reference for anyone who wishes to get up to speed with the latest developments in X-ray microcalorimeters.