The primary goal of the research project is to understand the cosmological evolution of the universe, in particular its accelerating expansion, in the context of gravity theories that extend the framework of general relativity. These theories offer a potential explanation of the accelerating expansion by gravitational effects only. This approach has the advantage that the introduction of a new and unknown matter type called dark energy, which is used to explain the accelerating expansion in the standard model of cosmology, is not necessary. As an additional benefit, these theories may provide a potential purely gravitational explanation of dark matter effects as well. A particular theory which features an accelerating cosmological expansion and which will be investigated further during the research project is multimetric gravity. The matter content of this theory is given by N >= 3 copies of the well-known standard model, each of which couples to its own metric tensor. The only interaction between the different standard model copies is the gravitational interaction mediated by an interaction between the N metrics. Theories of this type have been shown to be consistent with solar system observations at the post-Newtonian level. Further theories which will be examined include area-metric gravity and Finsler gravity. These theories extend the theory of general relativity by replacing the metric with a more general geometric structure of spacetime. A large fraction of the proposed research will be phenomenological. Particular aspects of the aforementioned theories which will be studied are the cosmological dynamics, structure formation, gravitational lensing, the deviation from Newtonian gravity on solar system scales, and the production and propagation of gravitational waves. These topics are closely connected to current experiments in the fields of cosmology, high-precision measurements of gravity and gravitational wave physics.