How giant planets form is one of the major open issues in planetary science. Given that gas giants gravitationally dominate their planetary systems, this limits our understanding of the origin of planetary system architectures and the conditions needed for the formation of habitable terrestrial planets. The standard model of giant planet accretion (Pollack et al 1996) consists of two phases: the accretion of a solid rocky/icy core of several Earth masses and the formation of a massive atmosphere around such a core by the capture of gas from the circumstellar disk. However, each of these phases presents significant unsolved problems. First, the accretion of cores from a population of planetesimals seems to stall long before a mass of several Earth masses is reached. Second, simulations predict that the capture of gas by a massive core, once it enters a runaway phase, does not stop until several Jupiter masses of gas are accreted (unless some fine-tuning of the removal of the circumstellar gaseous disks is invoked), in clear conflict with the masses of Jupiter and Saturn and the population of giant extra-solar planets (in which Jovian and sub-Jovian planets are very abundant). With MOJO, we propose to carefully study these phases by combining comprehensive hydrodynamical simulations and N-body simulations of interacting planetary embryos embedded in circumstellar disks. We will test two promising new mechanisms for core accretion: (i) rapid formation of planetary embryos via the accretion of pebbles by the largest planetesimals and (ii) mutual accretion of the embryos due to convergent migration. The latter requires a realistic model of non-isothermal protoplanetary disks in which planetary embryos migrate via tidal interactions with the gas towards an equilibrium orbital radius. This equilibrium radius could be a sweet spot for the mutual accretion of embryos, leading to the formation of a few massive cores. Once embryos are massive enough to become giant planets, accretion of their envelope will be studied with a nested mesh code that can resolve the accretion flow and the circumplanetary disk. Recent results show that this disk should be much less ionized and viscous than the surface layers of the circumstellar disk. A low-viscosity circumplanetary disk may act as a bottleneck for gas accretion onto the planet, preventing the fast runaway phase. This process has the potential of increasing the gas accretion timescale, making it comparable with the disk lifetime. The diversity of giant planet masses may stem naturally from the similarities of these two timescales. Experience has taught us that the Solar System, with its numerous and precise observational constraints, is a key benchmark for formation models. Requiring exoplanet-oriented models to also match the Solar System has led to the identification of essential physical mechanisms that would have otherwise been missed. MOJO will thus focus first on giant planet formation in our Solar System. The study will then be applied to exoplanetary systems, for which constraints on both the giant planet and terrestrial planet populations are becoming more and more detailed thanks to numerous ground-based and space-based surveys. We will also study the consequences of the migration of the planetary embryos on the formation and survival of habitable terrestrial planets. Last but not least, images of protoplanetary disks with very high resolution (less than 10AU) are becoming possible with instruments such as ALMA and SPHERE. Our disk-planet models can be used to interpret these observations. With MOJO, we have assembled a team with a range of expertise necessary to develop a model that explains both the giant planets of our Solar System and the diversity of planetary systems.