Clusters of galaxies at intermediate redshifts (0.2 <~ z <~ 0.8) are important cosmological probes because they trace the largest matter overdensities in the Universe. Thus cluster's masses as well as their evolution in numbers and temperature offer sensitive probes of the Universal matter density (in both baryons and dark matter). Indeed, rich clusters of galaxies were the first indicators that the cosmological matter density is less than the critical amount necessary to halt the Universal expansion. Therefore, the continued study of these important cosmological objects is warranted. In this dissertation I have conducted three studies related to the cosmological parameters using intermediate redshift clusters of galaxies from the Einstein Extended Medium Sensitivity Survey (EMSS), the most intensively studied sample of distant clusters: (1)I have studied the incompleteness of the EMSS cluster sample by using a new detection algorithm designed to find cluster X-ray sources which may have been missed by the EMSS X-ray detection algorithm. To determine the nature of these new sources, I have conducted an extensive campaign of optical imaging, and literature and database searches. I have found several rich, distant (z > 0.14), and X-ray luminous (log LX > 44 ergs s-1), clusters missed by the EMSS. Based upon these new discoveries, I find that the EMSS cluster sample is 73-82% complete. The addition of these new clusters reduces the measured evolution in the X-ray luminosity function (XLF) to <=1σ. From my optical campaign, I have determined the new clusters to be similar in richness and galaxy populations to other X- ray detected clusters in the EMSS. (2)I have used Röntgen Satellite (ROSAT) X-ray images to measure the X-ray sizes and shapes of EMSS clusters, thus reducing important systematic errors in calculating the XLF for these clusters. My measurements of parameters describing the surface brightness profiles show large variations from canonically assumed values. I have also used available ASCA cluster temperature measurements for a more precise K-correction. Including the addition of the newly discovered EMSS clusters, this results in a final XLF determination which shows no evolution out to z = 0.5. The combination of results from parts (1) and (2) requires that current values obtained from the EMSS survey of Ω matter = 0.45 +/- 0.1 for an open universe, and Ω matter = 0.29 +/- 0.1 for a flat universe must be considered upper limits. These results also highlight the sensitivity of XLF evolution studies to assumed cluster morphology, as well as the importance of surface brightness limits in establishing completeness of X-ray surveys for distant clusters. (3)I have investigated a subsample of the EMSS clusters to determine total virialized cluster masses from the X-ray data and used these results to compare with previously determined dynamical masses from optical studies of galaxy velocities and projected masses from weak-lensing signatures. I find no systematic bias between X-ray and dynamical methods across the sample, although individual clusters exhibit mass discrepancies up to a factor of 2. Weak gravitational lensing masses appear to be systematically higher than X-ray results by factors of ~50%, while strong lensing estimates show larger discrepancies (factors of ~2.5). X-ray derived cluster gas masses are calculated, from which I obtain a cluster baryon fraction of ~5% h- 3/2100 , yielding Ω0 ~ 0.3 h- 1/2100 .