This dissertation is concerned with extensions and wedge products of positive currents. Our study can be considered as a generalization for classical works done earlier in this field. Paper I deals with the extension of positive currents across different types of sets. For closed complete pluripolar obstacles, we show the existence of such extensions. To do so, further Hausdorff dimension conditions are required. Moreover, we study the case when these obstacles are zero sets of strictly k-convex functions. In Paper II, we discuss the wedge product of positive pluriharmonic (resp. plurisubharmonic) current of bidimension (p,p) with the Monge-Ampère operator of plurisubharmonic function. In the first part of the paper, we define this product when the locus points of the plurisubharmonic function are located in a (2p-2)-dimensional closed set (resp. (2p-4)-dimensional sets), in the sense of Hartogs. The second part treats the case when these locus points are contained in a compact complete pluripolar sets and p≥2 (resp. p≥3). Paper III studies the extendability of negative S-plurisubharmonic current of bidimension (p,p) across a (2p-2)-dimensional closed set. Using only the positivity of S, we show that such extensions exist in the case when these obstacles are complete pluripolar, as well as zero sets of C2-plurisubharmoinc functions. / At the time of doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Accepted. Paper 2: Manuscript. Paper 3: Manuscript.

Cette thèse comporte deux parties: - Dans la première partie, nous traitons d'abord une version kahlérienne du célèbre théorème d'extension d'Ohsawa-Takegoshi, puis, un problème de prolongement des courants positifs fermés. Notre motivation provient de la conjecture de Siu sur l'invariance des plurigenres dans le cas d'une famille kahlérienne. En effet, dans la preuve du célèbre théorème d'invariance des plurigenres de Siu, le théorème d'extension d'Ohsawa-Takegoshi joue un rôle important. Il est donc naturel de penser que la preuve de la conjecture fera également intervenir un théorème d'extension de type Ohsawa-Takegoshi dans le cas kahlérien. Suite aux difficultés techniques qui proviennent de la régularisation des fonctions quasi-psh sur les variétés kahlériennes compactes, nous obtenons seulement deux cas particuliers du résultat espéré. Pour ce qui est du prolongement des courants positifs fermés, notre résultat est un cas particulier de la conjecture qui prédit que tout courant positif fermé défini sur le fibré central d'une classe de cohomologie kahlérienne tordue par la classe de Chern du fibré canonique admet un prolongement. - Dans la deuxième partie, nous nous intéressons à l'unicité des solutions des équations de type Monge-Ampère généralisées. Il s'agit d'une généralisation d'un théorème de Bando-Mabuchi concernant les métriques de Kahler-Einstein sur les variétés de Fano. Nous suivons la méthode introduite par Berndtsson et généralisons son résultat en travaillant avec un courant positif fermé à la place d'une paire klt dans son contexte. Les propriétés de convexité des métriques de Bergman jouent un rôle important dans cette partie / This thesis consists in two parts: -In the first part, we first deal with a Kahler version of the famous Ohsawa-Takegoshi extension theorem; then, a problem of extending the closed positive currents. Our motivation comes from the Siu's conjecture on the invariance of plurigenera over a Kahler family. Indeed, in the proof of his famous theorem, the Ohsawa-Takegoshi theorem plays an important role. It is, therefore, natural to think that the proof for the conjecture involves an extension theorem of Ohsawa-Takegoshi type in the Kahler case. Because of the technical difficulties coming from the regularization process of quasi-psh functions over the compact Kahler manifolds, we only obtain two special cases of the hoped result. As for the extension of closed positive currents, our result is a special case of the conjecture which predicts that every closed positive current defined over the central fiber in a Kahler cohomology class twisted by the first Chern class of the canonical bundle admits an extension. -In the second part, we are interested in the uniqueness of the solutions of the equations of generalized Monge-Ampère type, a generalized Bando-Mabuchi theorem concerning the Kahler-Einstein metrics over Fano manifolds. We follow the method introduced by Berndtsson and generalize his result by working with a closed positive current in place of a klt pair in his context. The properties of the convexity of the Bergman metrics play an important role in this part