The objective of this thesis is to use astronomical observations to constrain the parameters of the cosmological models in the Big Bang scenario. The astronomical objects used in this work to derive cosmological information include clusters of galaxies and Cepheid variable stars. Clusters of galaxies contain large amounts of hot gas which radiates in the X-ray wavelengths. I present our results of analysing X-ray data of several clusters obtained by ROSAT and ASCA satellites. The imaging data of ROSAT provides accurate information on the distribution of the baryonic matter in these clusters. The spatially resolved spectroscopy data of ASCA yields the distributions of the gas temperature. In the hydrostatic equilibrium scheme, we use this information to derive the distribution of the total, baryonic and dark matter in these clusters up to virial radii. Clusters form through the collapse of a large volume of primordial matter and thus the matter content and distribution inside clusters may closely represent the universal ones. Assuming this, we derived cosmological information from the mass distributions inside clusters which imply that the mass density of the Universe is below the critical and thus the cosmic expansion will continue forever. The Cepheids are very bright pulsating stars whose brightness varies periodically. The period length and the mean luminosity of the Cepheids are tightly correlated which provides a widely used tool for cosmic distance determination. Collecting data from the literature we examined the metallicity effect on this relation, which is usually not considered as a systematical effect. Requiring statistical consistency we were able to bring several intrinsically discrepant analyses into agreement, and thereby determined the magnitude of the metallicity effect, as well as the value for the Hubble constant which describes the cosmic expansion velocity at a given distance scale. Combining this information with our results of the cosmic mass density based on clusters, we get an estimate for the age of the Universe as 12-14 billion years.