Abstract : The elliptical instability takes place in any rotating fluid with elliptical streamlines. Its existence in geo- or astrophysical flows raises many issues. This is the starting point of this theoretical, numerical and experimental work. After introducing basics of the rotating flows, chapter 1 presents the three considered planetary mechanical forcings: tides, precession and libration. Chapter 2 presents first numerical simulations of the elliptical instability in an ellipsoidal geometry. These simulations allow to derive the scaling laws needed to bridge the gap between numerics and planetary applications. Section 2.4 shows that the simultaneous presence of tides and libration can excite an elliptical instability inside synchronized celestial bodies. In section 2.5, a theoretical analysis of the interaction between tides and precession is developed and validated. Finally, in section 2.6, we prove that the elliptical instability can still develop over convective flows. Chapter 3 focuses on the magnetohydrodynamics of the elliptical instability. New results on the magnetic induction by the elliptical instability are obtained and validated numerically. An experimental work, based on a MHD setup, is then described. Our measurements allow to study the dynamics of the instability under an external imposed magnetic field. The experimental setup is then modified to set up a synthetic dynamo. Chapter 4 study the presence of the elliptical instability in known planets, moons, and stars. The particular case of the Moon is first considered and a scenario, based on the elliptical instability, is proposed and evaluated to explain the early lunar dynamo. Telluric bodies are then considered, and an adapted stability analysis shows that the instability can be expected in the Early Earth, Europa and three exoplanets (55CnCe, CoRoT-7b et GJ1214b). Finally, the possible development of the instability in extra-solar Hot-Jupiters systems is considered, showing its relevance for some of them, such as the system of Tau-boo.