This thesis work focuses on the understanding of the formation and subsequent evolution of the population of elliptical galaxies in rich galaxy clusters. Elliptical galaxies define a unique and simple laboratory to study the process of galaxy formation and the evolution of stellar populations. Observational data, including a sample of eight clusters observed at Cerro Tololo Inter-American Observatory (CTIO) and ten clusters observed with the Hubble Space Telescope (HST), have been used to conduct several related studies. A surface brightness analysis of the ellipticals in the cores of ten galaxy clusters observed with HST shows that elliptical galaxies are well-represented by an 'r'1/4-law light profile and follow a well-defined relation in the magnitude/size plane, up to 'z' = 1.2. This relation shifts, on average, towards higher luminosities (or equivalently, to smaller sizes) with increasing redshift. If we assume that the size of these galaxies are fixed during the period from 'z' = 1.2 to the present, then these galaxies are evolving monotonically in luminosity. This luminosity evolution amounts to [Delta]'MB' = -1.2 at 'z' = 1.2, when compared to the galaxies in local clusters, and is consistent with that expected for an old and passively evolving stellar population. The luminosity functions for the red galaxies from the eight clusters in the CTIO sample show that are well-represented by a Schechter function, consistent with a single value for their characteristic magnitude (M*). When compared to local cluster luminosity functions, there is evidence for a brightening of a few tenths of a magnitude in M* (the exact value depends on the cosmology). For a low density universe, the brightening observed in the luminosity function, up to 'z' = 0.45, is consistent with that observed from the surface brightness analysis. The cluster elliptical galaxies from the CTIO sample, selected morphologically using a galaxy light profile fitting technique, define a very tight sequence in the colour/magnitude plane. All the clusters are consistent with having a single value for their intrinsic colour dispersions, [delta] = 0.06 ± 0.02, in the restframe ('U' - 'V')/' V' colour relation. These small colour dispersions imply the stars in these galaxies have a typical age dispersion from galaxy to galaxy of ~20% of their age at 'z' = 0.45. Similarly, when comparing the mean colour for the different clusters, it was found that they are consistent with having a single mean colour, ('V' - 'I') = 2.055 ± 0.028. The small dispersion in the mean colours is completely accounted for by sampling uncertainties. With the addition of 'K'-band photometry, and data for local ellipticals, it has been possible to determine the evolutionary vector in the restframe ('U' - 'V')/(' V' - 'H') colour plane from 'z' = 0.45 to 'z' = 0. Predictions of current models have been tested against this observed vector. Most of the more popular models have failed to account for this colour change, and only a more sophisticated model--including infall of primordial gas and chemical enrichment during the assembly of the stellar population--was able to reproduce the observed evolution. The observations in this work strongly suggest that these galaxies are old systems formed at high redshift. However, given the quantitative uncertainties in the observations, population synthesis models, cosmology, and uncertainties in how much synchronicity in formation is reasonable, the observations do not rule out scenarios involving formation at lower redshifts.