This will be an introductory talk, whereby I will speak on the mathematics of the modern theory of cosmology. This theory is based on Einsteinšs gravitational field equations for a perfect fluid, together with astronomical observations. Both of these topics will be considered, and I will also discuss the very interesting "standard" cosmological model. Towards the end of my talk, I will explain the recent modification of this theory, due to myself and Blake Temple, (University of California at Davis). The standard model, predicts a Universe of infinite mass and extent at each instant after the Big-Bang. We propose a modification of the standard model by one based on the mathematical theory of shock waves, which avoids this defect. In order to get the shock wave to lie beyond one Hubble length at present time, (this is needed to agree with astronomical observations), we prove that the expansion must take place inside a black hole (really a "white hole"). There are other interesting and unexpected features of our model, which will be discussed. I will also explain the undefined notions mentioned above.