In the first place I have reviewed the true foundation of Einstein's theory of general relativity, the so-called principle of equivalence, according to which there is no essential difference between "genuine" gravitation and inertial forces, well known from accelerated vehicles. By means of a comparison with Gaussian geometry of curved surfaces-the background of Riemannian geometry, the tool used by Einstein for the mathematical formulation of his theory-it is made clear that this principle is incompatible with the idea proposed by Mach and accepted by Einstein as an incitement to his attempt to describe the main situation in the universe as an analogy in three dimensions to the closed surface of a sphere. In the later attempts toward a mathematical description of the universe, where Einstein's cosmology was adapted to the discovery by Hubble that its observed part is expanding, the socalled cosmological postulate has been used as a kind of axiomatic background which, when analyzed, makes it probable that this expansion is shared by a very big, but still bounded system. This implies that our expanding metagalaxy is probably just one of a type of stellar objects in different phases of evolution, some expanding and some contracting. Some attempts toward the description of this evolution are sketched in the article with the hope that further investigation, theoretical and observational, may lead to an interesting advance in this part of astrophysics.