Active galactic nuclei (AGNs) are ideal laboratories for fundamental physics and cosmology. Our knowledge of AGNs has been greatly advanced during the past decades thanks to the unprecedentedly powerful X-ray missions (e.g., Chandra, XMM-Newton, and Swift) and modern large area surveys in the optical/UV band (e.g., the Sloan Digital Sky Survey; SDSS). Studies on the relation between multiwavelength properties of AGNs can reveal the physics of AGN accretion process. In this dissertation, I mainly study the relation between the AGN X-ray properties and their optical/UV spectral properties on emission lines, absorption lines and continuum. I also investigate the X-ray properties of AGNs with extreme radio properties in the early Universe. (1) We studied the X-ray properties of a class of radio-quiet SDSS quasars with weak broad emission lines (weak-line quasars; WLQs). Although the WLQ population shows diverse X-ray properties, they have an excess of X-ray weak sources. Besides having weak emission lines, the X-ray weak WLQs generally show other unusual UV emission-line properties similar to those of the remarkable X-ray weak quasar PHL 1811 (e.g., highly blueshifted C IV lines, weak semi-forbidden lines, and strong UV Fe emission). They are classified as "PHL 1811 analogs". The X-ray weak WLQs also show a harder X-ray spectrum, while the WLQ with normal X-ray brightness have similar X-ray spectral properties to those of typical quasars. We proposed an AGN geometry which can potentially unify the X-ray weak and X-ray normal WLQ populations via orientation effect. The infrared-to-UV spectral energy distributions (SEDs) of X-ray weak and X-ray normal WLQs have consistent SEDs with those for typical quasars, which disfavors the BL Lac-like scenario for the nature of these quasars (Chapters 2 and 3). (2) We have led the best X-ray study to date on quasars with intermediate-width absorption lines (mini-BALs). We found the X-ray brightness of mini-BAL quasars are more close to those of typical quasars than to BAL quasars (which are generally X-ray weak), showing they do not have substantial X-ray absorption. Strong correlations were found between the X-ray brightness and UV absorption parameters, e.g., the absorption strength and maximum outflow velocity. We further proposed new UV absorption parameters which better correlate with the X-ray properties than existing parameters do (Chapter 4). (3) We studied the relation between the optical/UV luminosity and X-ray luminosity (quantified by the alpha_ox parameter) for the most-luminous quasars over a wide range of redshifts. Our correlation analyses provide better constraints on the alpha_ox-UV luminosity correlation. We have also verified that the alpha_ox parameter does not significantly evolve with redshift. We provide the individual and composite mid-infrared-to-UV SEDs for the most-luminous quasars. (Chapter 5). (4) We presented the X-ray and multiwavelength properties of the highly radio-loud quasars (HRLQs) at z > 4. Our HRLQs show a significant enhancement of X-ray emission over those HRLQs at lower redshift with similar optical/UV and radio luminosities, suggesting that the jet-linked X-ray emission mechanism in the early universe may differ from that in the more evolved universe. The optical/UV emission-line strength of RLQs are correlated with radio loudness, but not with relative X-ray brightness. Our HRLQs generally follow the anti-correlation between radio loudness and X-ray power-law photon index. We also studied the broad-band SEDs of HRLQs. Some HRLQs have an excess of mid-infrared emission which may originate from the jet synchrotron emission. None of our z > 4 HRLQs is detected by the Fermi LAT two-year survey (Chapter 6).