Since the seminal work of Bardeen-Cooper and Schrieffer (BCS) that unveiled the microscopic mechanisms of superconductivity, it has been known that in fermionic ensembles superfluidity could arise through Cooper pairing of particles with opposite spins. Two extensions of their works were quickly proposed: first, was there a relationship between BCS superfluidity and Bose-Einstein condensation, and second what is the fate of the superfluid state when the populations of the spin states are imbalanced.
After decades of theoretical investigations, the experimental answer to these two questions was given by a series of experiments performed in the past few years on ultra cold fermions. We will present experimental results obtained at ENS confirming the scenario of a smooth crossover between the BCS state and a Bose Einstein condensate of deeply bound molecules. We will also present a theoretical investigation of experiments performed at Rice and MIT on the influence of spin imbalance on fermionic superfluidity. They both confirm the robustness of the BCS state, and we will show that the most salient features can be quantitatively recovered by the study of the polaron problem, ie the behavior of an impurity immersed in a Fermi Sea.