The "Dark Energy and Massive Neutrino Universe" project is a set of 14 large N-body simulations to study the evolution of large scale structures in the presence of massive neutrinos and dynamical dark-energy, parametrised according to Chevallier, Polarski & Linder. As shown in recent studies, the effect of the neutrino mass on the large scale structure of the Universe (growth rate and BAO) can degrade constraints on the dark energy equation of state, undermining the interpretation of cosmological observations. Non-linear modelling of structure formation is therefore needed to fully exploit the current and future very large cosmological data sets. These simulations are the first in the literature to simultaneously deal with a different dark-energy equations of state and massive neutrinos as a collisionless particle component. The simulations are characterised by a box side L=2 Gpc/h, a particle number of N=2x(2048)^3 (the factor of 2 stands for CDM and neutrino particles), and a mass resolution for CDM particles of about 8x10^10 M_sun/h. They have been produced via a modified version of the GADGET-3 code which includes massive neutrinos as a particle component, and accounts for quintessence models in the background evolution. Given their large volume and mass resolution, the DEMNUni simulations are being used to generate mock galaxy/void catalogues, CMB/weak-lensing/SZ/ISW/Rees-Sciama maps, mimicking the observations of future galaxy surveys and CMB probes. This will provide an important contribution to the preparation of future cosmological probes, which aim at measuring the dark-energy equation of state and the total neutrino mass with unprecedented accuracy. I will present several results obtained from the ongoing analysis of the DEMNUni simulations.