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In this thesis, we have explored what information may be gleaned from X-ray observations of galaxies in dense environments. We use X-ray observations from XMM- Newton and the Chandra X-ray Observatory, and multi-wavelength ancillary data, to investigate the X-ray emission of galaxies. First, we study the distribution and properties of the intragroup diffuse X-ray emission in compact groups (CGs) of galaxies. From a sample of 19 CGs, we find the morphology of hot gas in low-mass groups is varied, and most systems have hot gas (if any) associated with only individual members. The galaxy-linked hot gas is coupled with high star formation rates (SFRs), while only CGs with high baryonic masses have substantial hot gas linked to the group environment. It is high-mass CGs that also agree well with the observed scaling relations between diffuse X-ray luminosity (LX), gas temperature, and velocity dispersion predicted and observed in galaxy clusters, indicating that the hot gas in only massive CGs is virialized. We also investigate the relations between LX, SFR, and stellar mass from individual members of CGs and the infall region of the nearby Coma galaxy cluster, which is the only environment that has a mid-infrared galaxy color distribution similar to CGs. The Coma galaxies agree with the scaling relations between LX, SFR, and stellar mass from the literature within uncertainties, while the CG members often show an X-ray excess. We also used our multi-wavelength observations to identify active galaxies in the Coma infall sample and find that the fraction of active galaxies is similar to the CG environment. From our observations of the diffuse X-ray emission in CGs, we find it unlikely that the intragroup hot gas is responsible for the rapid transformation of galaxies from star-forming to quiescent. While the fraction of nuclear activity in Coma infall and CG galaxies is similar, which may reflect the influence of multi-galaxy gravitational interactions, the X-ray emission from individual galaxies in the two environments is also markedly different.