In this talk, we will present recent experimental work performed at ENS on the thermodynamics of strongly interacting 6Li Fermi gases. We have developed a general method to probe with high precision the thermodynamics of locally homogeneous ultracold gases [1]. This allows stringent tests of recent many-body theories. First, we focus on the finite-temperature Equation of State (EoS) of the unpolarized unitary gas. Precise thermometry is provided by adding to the trapped 6Li unitary gas a small amount of bosonic 7Li that is weakly perturbing the system. We show that the low-temperature properties of the strongly interacting normal phase are well described by Fermi liquid theory and we localize the superfluid phase transition. Second, we address the zero-temperature EoS of the spin-polarized system. Surprisingly, despite strong interactions, the normal polarized phase behaves as a mixture of two ideal gases: a Fermi gas of bare majority atoms and a non-interacting gas of dressed quasi-particles, the Fermi polarons. Finally, we will report on the extension of our study to the BEC-BCS crossover in the low temperature limit [2]. A detailed comparison with theories including Monte-Carlo calculations and the Lee-Huang-Yang corrections for low-density bosonic and fermionic superfluids is presented. Our equation of state can be directly used to describe low density neutron matter such as the outer shell of neutron stars. [1] S. Nascimbène, N. Navon, K. Jiang, F. Chevy, and C. Salomon, Nature 463, 1057 (2010) [2] N. Navon, S. Nascimbène, F. Chevy, and C. Salomon, Science Express, published on line April 15, 2010