This thesis presents some results of the theory of noncommutative probability. It deals in particular with martingale inequalities in von Neumann algebras, and their associated Hardy spaces. The first part proves a noncommutative analogue of the Davis decomposition, involving the square function. The usual arguments using stopping times in the commutative case are no longer valid in this setting, and the proof is based on a dual approach. The second main result of this part determines the dual of the conditioned Hardy space h_1(M). These results are then extended to the case 1<p<2. The second part proves that an atomic decomposition for the Hardy spaces h_1(M) and H_1(M) is valid for noncommutative martingales. Interpolation results between the spaces h_p(M) and bmo(M) are also established, with respect to both complex and real interpolations. The two first parts concern discrete filtrations. In the third part, we introduce Hardy spaces of noncommutative martingales with respect to a continuous filtration. The analogues of the Burkholder/Gundy and Burkholder/Rosenthal inequalities are obtained in this setting. The Fefferman-Stein duality and the Davis decomposition are also successfully transferred to this situation. The proofs are based on ultraproduct techniques and L_p-modules. A discussion about a decomposition involving algebraic atoms gives the expected interpolation results
