Normal galaxies are galaxies whose emission is not dominated by active galactic nuclei (AGN). Current X-ray observatories allow X-ray studies of normal galaxies beyond the local group (≥ 1 Mpc). This thesis presents a study of normal galaxy samples drawn from serendipitous sources detected by the XMM-Newton Observatory up to a few hundred Mpc.\ud The work begins with a pilot study in which a sample of 72 normal galaxy candidates is selected from the XMM-Newton/2dF Wide Angle Serendipitous Survey using X-ray-to-optical flux ratios, X-ray luminosities and the 2dF optical spectra. This sample is classified, based on optical emission lines, into five subsamples: AGN, star-forming (SF) galaxies, composite galaxies, unclassified narrow-emission-line galaxies (NELGs) and absorption-line galaxies (ALGs). The X-ray properties of these subsamples, shown by X-ray spectra and hardness ratios, are broadly consistent with the optical classifications. \ud The Second XMM-Newton Serendipitous Catalogue (2XMM) is then used to cross-correlate with Sloan Digital Sky Survey (SDSS) and yields a sample 463 normal galaxy candidates (the XS sample) using the same selection criteria as in the pilot study. Using the continuum-subtracted optical emission line measurements from the Max-Planck-Institute for Astrophysics/Johns Hopkins University catalogues (the MPA/JHU catalogues), a complete subset of the XS sample is separated into five subsamples: AGN, SF galaxies, composite galaxies, unclassified NELGs and ALGs. The X-ray properties, on the basis of the X-ray spectra and hardness ratios, are largely consistent with expectations from their optical classifications. Additional parameters from the MPA/JHU catalogues, such as star-formation rates (SFRs) and stellar masses, are used for further investigation of the nature and properties of individual subsamples. The effectiveness of the normal galaxy selection criteria is investigated to show how efficient the X-ray-to-optical flux ratios are on separating AGN from truly normal galaxies
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