"One of the most challenging tasks for theoretical and observational cosmology in the next decade will be to understand the fundamental nature and properties of Dark Energy and Dark Matter. To this end, several ambitious observational programs are presently ongoing or have been planned for the near future, with the aim to reach unprecedented accuracy in the investigation of the dark sector of the Universe. The large wealth of high-quality data expected from these wide surveys will however require equally reliable and accurate theoretical predictions for different and competing cosmological scenarios in order to exploit at best their potential discriminating power.Such predictions often need to rely on large numerical simulations of structure formation which require sophisticated algorithms and the use of large supercomputing systems. However, while suitable simulation techniques have now been firmly established for the standard cosmological model, new algorithms and new simulations are needed in order to provide a comparable set of predictions for a significant number of alternative and competing scenarios. The present proposed research program aims at developing suitable numerical codes and perform large simulations for a wide range of non-standard cosmological models, ranging from dynamical Dark Energy scenarios, to Modified Gravity Theories, to alternative Dark Matter particle candidates or different types of primordial non-Gaussianity. Without such effort the discriminating power of any observational enterprise would not be exploited at best and might fail to shed light on the Dark Universe. The present proposed research plan is therefore an essential step to provide a direct link between theory and observations in cosmology, with a particular relevance for the success of present and future observational programs of the European scientific community."