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ABSTRACT: In this thesis the cosmological constant is investigated from two points of view. First, we study the influence of a time-dependent cosmological constant on the late-time expansion of the universe. Thereby, we consider several combinations of scaling laws motivated by renormalisation group running and different choices for the interpretation of the renormalisation scale. Apart from well known solutions like de Sitter final states we also observe the appearance of future singularities. As the second topic we explore vacuum energy in the context of discrete extra dimensions, and we calculate the Casimir energy density as a contribution to the cosmological constant. The results are applied in a deconstruction scenario, where we propose a method to determine the zero-point energy of quantum fields in four dimensions. In a related way we find a lower bound on the size of a discrete gravitational extra dimension, and finally we discuss the graviton and fermion mass spectra in a scenario, where the extra dimensions form a discrete curved disk.