The prime scientific objectives of the NASA Small Explorer mission, Gravity and Extreme Magnetism SMEX, or “GEMS”, are to determine the effects of the spin of black holes, the configurations of the magnetic fields of magnetars, and the structure of the supernova shocks which accelerate cosmic rays. In the cases of both stellar black holes and supermassive black holes, sensitivity to 1% polarization is needed to make diagnostic measurements of the net polarizations predicted for probable disk and corona models. GEMS can reach this goal for several Seyferts and quasars and measure the polarizations of representatives of a variety of other classes of X-ray sources, such as rotation-powered and accretion-powered pulsars. GEMS uses foil mirrors to maximize the collecting area achievable within the SMEX constraints. The polarimeters at the mirror foci are time projection chambers which use the photoelectic effect to measure the polarization of the incident photon. We have built laboratory models with good efficiency and modulation in the 2–10 keV range. An attached small student experiment would add 0.5 keV sensitivity for bright, soft sources. The instrument has a point spread function which allows measurement of structures in the brighter nearby supernova remnants. GEMS' Orbital Sciences spacecraft will rotate at a rate of 0.1 revolutions per minute during observations, so that systematic errors due to the detector can be detected and corrected. A program of 35 sources can be observed in 9 months. […]