Aspects of cosmology and quantum gravity in an accelerating universe

The observation that we are living in a Universe that is expanding at an ever-increasing rate is a major challenge for any fundamental theory. The most obvious explanation for an accelerating Universe is a positive cosmological constant, but we do not really know how to do quantum field theory or string theory in spacetimes that are not asymptotically flat. In this thesis, we address various issues that arise in this general context. The problems we address include the stability and evolution of de Sitter-like compactifications, the possibility of defining a quantum theory in de Sitter space using quantum groups, and finally, the classical evolution of thin shells (boundaries of new phase bubbles) in an inhomogeneous Universe with positive Λ.