We provide a concise review of X-ray observations of synchrotron and inverse Compton radiation from relativistic electrons in cosmic sources, in the context of synergies between X-ray and γ-ray astronomy. Particular emphasis is placed on the cases of supernova remnants, pulsar wind nebulae, and relativistic jets of quasars. We discuss that imaging spectroscopy of synchrotron X-ray emission plays key roles in studying acceleration and transport of high-energy electrons, as well as in probing the magnetic field through a comparison with TeV γ-ray data. To demonstrate some prospects for future X-ray observations, we showcase the scientific capabilities of the next major X-ray observatory, ASTRO-H, which is a joint JAXA-NASA mission to be launched in 2014.