In this paper we review cosmological relativity,a new special theory of relativity that was recentlydeveloped for cosmology, and discuss in detail some ofits aspects. We recall that in this theory it is assumed that gravitation is negligible.Under this assumption, the receding velocities ofgalaxies and the distances between them in the Hubbleexpansion are united into a four-dimensionalpseudo-Euclidean manifold, similarly to space and time inordinary special relativity. The Hubble law is assumedand is written in an invariant way that enables one toderive a four-dimensional transformation which issimilar to the Lorentz transformation. The parameter inthe new transformation is the ratio between the cosmictime to the Hubble time (in which the cosmic time ismeasured backward with respect to the present time). Accordingly, the new transformationrelates physical quantities at different cosmic times inthe limit of weak or negligible gravitation. Thetransformation is then applied to the problem of the expansion of the universe at the very earlystage when gravity was negligible and thus thetransformation is applicable. We calculate the ratio ofthe volumes of the universe at two different timesT₁ and T₂ after the big bang. Under theassumptions that T₂ – T₁ 10^-32 sec and T₂ 1 sec,we find that V₂/V₁ =10^-16/T₁. For T₁ 10^-132 sec we obtainV₂/V₁ 10⁵⁰. Thisresult conforms with the standard inflationary universe theory, but now it isobtained without assuming that the universe is propelledby antigravity. New applications of the theory arepresented. This includes a new law for the decay of radioactive materials that was recentlydeveloped by Carmeli and Malin. The new law is amodification of the standard exponential formula whencosmic times are considered instead of the ordinarylocal times. We also show that there is no need to assumethe existence of galaxy dark matter; the Tully-Fisherlaw is derived from our theory. A significant extensionof the theory to cosmology that was recently made by Krori, Pathak, Das, and Purkayastha isgiven. In this way cosmological relativity becomes ageneral theory of relativity in seven dimensions ofcurved space-time-velocity. The solutions of the field equations in seven dimensions obtained by Kroriet al. are given and compared to those of the standardFriedmann-Robertson-Walker result. A completely newpicture of the expanding universe is thus obtained and compared to the FRW one.