X-ray polarimetry is a prime tool for investigating the physics of compact objects, which has not been adequately exploited thus far. However, current low-cost technology and modest launch requirements could provide a large number of positive observations with a sensitivity factor at least 100 greater than 10 years ago. The amount of astrophysical information potentially to be gained from this is enormous. The introduction of polarimetric information (direction and degree) would bring a quantum jump in the parameter space used to investigate compact objects, from the current two (spectra and time variability) to four independent parameters that models need to satisfy. This should greatly improve our ability to discriminate between various possible models. Such observations could lead to an elucidation of the rotation-powered and accretion-powered pulsar radiation mechanisms, could help clinch the identification of black hole canditates, and could decide between thermal and nonthermal AGN radiation models, as well as pin down the geometry of the accretion flows in both galactic and extragalactic sources. 56 references.