Final Activity Report Summary - ExtraDim (Cosmological challenges in theories of extra dimensions) In this work we investigated theories according to which our universe has more than four dimensions. Liberating our universe from four dimensions offers new perspectives to long standing problems of high energy physics and cosmology. My research project focussed on three areas of interest in the cosmology of extra dimensions. Firstly, I studied the explanation of the smallness of the vacuum energy of the universe through extra dimensional theories. During the course of the fellowship, I gave specific examples of models which could accomplish that and developed techniques to consistently study them. This was done for a specific type of extra dimensional models, with our universe being a four dimensional surface in a six dimensional space. Furthermore, I developed the cosmology of such extra dimensional models. Secondly, I studied the cosmology of the moduli fields, corresponding to the sizes and shapes of extra dimensions. Using specific theories, I showed how some of them could be stabilised and how cosmological inflation could be realised using the modulus corresponding to the position of our universe within the extra dimensions. I also gave testable predictions for such models which could be confronted with future observations. Thirdly, I studied the extra dimensional models which predicted modifications of gravity at large distances, as an alternative to the dark energy paradigm. I showed that, in some extra dimension models, the current acceleration of the universe could be attributed to the influence of the extra dimensions instead of the usually introduced mysterious dark energy. Furthermore, I explored the cosmology of such extra dimensional models with potentially interesting observational signatures.