In this thesis we study the edge currents and the multi-band superconductivity in the unconventional superconductor Sr2RuO4. Numerous measurements have given strong support for a topologically non-trivial time-reversal symmetry breaking chiral p-wave state in this material. However, the spontaneous edge current expected for this order has eluded experimental detection. In this thesis, we present a general theoretical description of the edge currents in chiral superconductors. Our results elucidate the connection between the edge currents and the topological property of the chiral pairing. On this basis, we argue that superconducting gap anisotropy, combined with surface disorder, may provide an explanation for the absence of observable edge currents in Sr2RuO4. In addition, contrary to intuitive expectations, the integrated edge current is found to identically vanish for any non-p-wave chiral superconductor in the continuum limit-- a result which may be connected with the orbital angular momentum problem in chiral superfluids, such as the A phase of He-3. In lattice models, the integrated edge current may not vanish in non-p-wave superconductors but, in general, is substantially smaller compared to that of a simple chiral p-wave. In a separate study, we investigate the multi-band nature of the superconductivity in Sr2RuO4, via explicit microscopic calculations of the multi-band interactions. Our results indicate comparable pairing correlations on all of the bands and the existence of soft collective phase fluctuations--a Leggett mode. We also examine the possibility of alternative time-reversal symmetry breaking multi-band superconductivity which does not necessarily require chiral p-wave pairing. / Thesis / Doctor of Philosophy (PhD)