I have attempted to present the basic ideas behind many (but not all) of the current methods used to estimate various Cosmological parameters, especially the Hubble constant (H0), the curvature of the Universe () and its matter content (m). To do so I reviewed some basics of the standard, Robertson-Walker, Cosmology in order to highlight the interrelations of the Cosmological parameters and the way they affect the global dynamics of the Universe.
The results of a variety of different analyses, based on a multitude of data, point towards a concordance model, which is a flat, 0.7, with an inflationary spectral index n 1, h 0.72, b 0.04, t0 1.31 Gyr's. Nevertheless there are conceptual problems and open issues that may or may not prove to be daunting.
For example, the = 0.7, h = 0.7 model seems to have problems generating the correct power on galactic scales. Detailed numerical simulations of  show that this model has much more power on small scales to be reconciled with observations (see discussion in ). Furthermore, there is the fine tuning problem. Why does the energy density of the vacuum have a value like: ~ m ~ 1, implying that it dominates the Universe just NOW! An easy way out would be to invoke anthropic arguments , but it is too feeble a justification, especially since the different contributions to at the early phase transitions are 50 - 100 orders of magnitude larger than what is observed. One would have to invoke a fine-tuning in order for the different contributions to cancel out, but yet not completely! However, theoretical models are being developed in an attempt to alleviate such problems (cf. ), some by allowing to be a function of time (see  and references therein).
I would like to thank Spiros Cotsakis and Lefteris Papandonopulos for organizing this wonderful school and for their hospitality in the island of Pythagoras. I would also like to thank the students of the school for their interest, their stimulating discussions and their ...dancing abilities. Many thanks to Ed Chapin that had the patience to go through the text and correct my "Greekisms".