EMERGIANT is a research project about Jupiter and Saturn, which lies at the interface between astrophysics and atmospheric science. EMERGIANT is fundamental research, driven by the incentive to learn more about the world we live in (the overarching goal of the ANR 2017 Défi des autres savoirs). The extreme weather of Jupiter and Saturn, our Solar System's so-called gas giants, is a permanent source of wonder and inspiring challenges. The community has been debatting for decades the mechanisms underlying fast-rotating gas giants' spectacular banded jets (strong planetary-scale wind currents), long-lasting traveling storms (e.g. centuries-old Jupiter's Great Red Spot), and powerful convective thunderstorms (e.g. Saturn's Great White Spots). Enigmatic differences between Jupiter and Saturn are still yet to be explained: although Saturn receives less sunlight and generates less internal heat than Jupiter, its equatorial super-rotating jet is three times stronger; and contrary to Jupiter, Saturn exhibits a remarkable hexagonal-shaped circumpolar jet. We are currently living an observational golden age for gas giants, with a wealth of observations on a variety of atmospheric phenomena in Jupiter and Saturn's troposphere and stratosphere -- raising new constraints and questions to achieve the overarching goal of fully understanding Jupiter and Saturn's atmosphere and climate. We propose with the EMERGIANT project to perform numerical atmospheric modeling on French supercomputers to make a paradigm shift from the phenomenological point of view imposed by observations, to a complete physically-based understanding of gas giants' atmospheres. We will carry out unprecedented dynamical simulations with a full Global Climate Model for gas giants to draw the big picture of gas giants' climate and meteorology. Not only this is a unique methodology to draw a comparative planetology approach between Jupiter and Saturn, but this will open broader perspectives of geophysical fluid dynamics' questions also valid on the Earth: the driving of jets by eddies from an inverse energy cascade, the emergence of cyclones / anticyclones, the impact of moist convection on the large-scale planetary circulations, and the stratospheric oscillations reminiscent of the Quasi-Biennal Oscillation on Earth. EMERGIANT will allow the scientific coordinator to build a strong local team at Laboratoire de Météorologie Dynamique / Université Pierre et Marie Curie, which will evolve in a high-level network of national, European, and international collaborations. The EMERGIANT project will enhance the scientific return of European and international space missions, as well as High Performance Computing capabilities in France and Europe. The simulation results obtained with the EMERGIANT gas giants Global Climate Model will be made available through an online graphical interface opened to scientific communities, students and the general public.