The use of microcalorimeters for high-resolution, high quantum efficiency, non-dispersive X-ray spectroscopy has been demonstrated over the past few years. During this time the energy resolution in the 1-10 keV band has improved from a value of-140 eV (FWHM), comparable to the best solid-state (Si/Li) detectors, to a value of-17 eV. We are presently working on improving the energy resolution by designing devices that have been optimized with respect to heat capacity, thermal conductance, X-ray absorbing material, thermistor geometry and thermistor implant concentration. In addition we have been working on the design of an integrated spectrometer consisting of a He4 cryostat containing an adiabatic demagnetization refrigerator, non-X-ray blocking filters and control and signal processing electronics. The essential components for such a system have been assembled at the University of Wisconsin and a similar system is being developed at Goddard. These systems will demonstrate the feasibility of this approach for space flight use, in particular for the NASA AXAF mission, and serve as research tools for further work in X-ray calorimetry. In this paper we give an overview of X-ray calorimetry, present the results of on-going X-ray tests, and discuss an approach for building an X-ray calorimeter spectrometer.