Many X-ray observations of GigaHertz Peaked Spectrum and Compact Steep Spectrum sources have been made with Chandra X-ray Observatory and XMM-Newton Observatory over the last few years. The X-ray spectra contribute the important information to the total energy distribution of the compact radio sources. In addition the spatial resolution of Chandra allows for studies of the X-ray morphology of these sources on arcsec scales and provide a direct view of their environments. This paper gives a review of the current status of the X-ray observations and their contribution to our understanding of the nature of these compact radio sources. It also describes primary physical processes that lead to the observed X-ray emission and summarize X-ray emission properties expected from interactions between an expanding radio source and the intergalactic environment. ; Comment: 6 pages, 9 figures, to appear in the proceedings of the '4th Workshop on Compact Steep Spectrum and Gigahertz-Peaked Spectrum Radio Sources', AN in press