In this dissertation, the generic, non-rotating, homogeneous closed model universe ( the "Mixmaster Universe", Bianchi Type IX) is studied to gain some insight into how the broad-scale homogeneity of the universe may have been produced at very early times. We begin our discussion by sketching the development of relativistic cosmology until the last decade. In the second chapter we discuss particle horizons in the Robertson-Walker models. These standard models of the universe possess particle horizons. Thus, only a finite part of such a universe could have been causally connected; while the isotropy of 2.7°K microwave radiation implies the universe to be homogeneous on a much larger scale than the size of the horizon. The third chapter discusses in detail the evolution of the Mixmaster Universe near the singularity using the Hamiltonian techniques developed by Misner for these models . At a fixed time (or volume) epoch Ω0, a Mixmaster Universe is specified by initial conditions' β+, β- (shape anisotropy) and p+ , p- (expansion rate anisotropy). In the fourth chapter we derive the equations for rays of high-frequency sound waves and light waves. When these equations are applied in the Mixmaster Universe, we find that for certain subsets of initial conditions, some of these sound rays and light rays would circumnavigate the corresponding universes in certain directions. Our results for light rays parallel those of Doroshkevich and Novikov, however we use entirely different methods (Hamiltonian methods) for treating the Einstein equations. In the last chapter the evolution of the Mixmaster Universe is shown equivalent to a geodesic flow within a bounded region of the Lobatchewsky plane. The boundary shape makes this flow Ergodic. The ergodicity is proved by invoking a certain group of conformal transformations, G, which makes this flow of broken geodesics on the Lobatchewsky plane, D, into a continuous one on D/G. The Einstein equations in this problem lead to a natural measure on initial conditions related to β+, p+. The measure of the circumnavigation sets depends upon the epoch and it goes to zero as the volume of the universe shrinks to zero. Finally, we compute the probability for circumnavigation along any one axis of the universe, It turns out to be roughly 1% for an empty universe and it decreases to 0.02% for realistic models containing radiation and matter in them.