This paper introduces the concept of barycenters in the Wasserstein space, generalizing McCann's interpolation to more than two measures. The authors provide existence, uniqueness, characterizations, and regularity results for the barycenter, relating it to the multi-marginal optimal transport problem. They also discuss examples, including a rigorous solution to the Gaussian case and an $L^\infty$ regularity result for the barycenter. The paper establishes a duality relationship between the primal and dual problems, leading to a characterization of the barycenter in terms of the Gangbo-Święch maps. The authors further explore the convexity properties of functionals in the Wasserstein space.This paper introduces the concept of barycenters in the Wasserstein space, generalizing McCann's interpolation to more than two measures. The authors provide existence, uniqueness, characterizations, and regularity results for the barycenter, relating it to the multi-marginal optimal transport problem. They also discuss examples, including a rigorous solution to the Gaussian case and an $L^\infty$ regularity result for the barycenter. The paper establishes a duality relationship between the primal and dual problems, leading to a characterization of the barycenter in terms of the Gangbo-Święch maps. The authors further explore the convexity properties of functionals in the Wasserstein space.