The optimal transportation problem originally introduced by G. Monge in 1781 and rediscovered by L. Kantorovich in 1942 consists in transformation of one mass distribution to another with the minimal amount of work. In this thesis, we consider some variational problems involving optimal transport. We are mainly motivated by the Wasserstein barycenter problem introduced by M. Agueh and G. Carlier in 2011, which provides a nonlinear averaging of probability measures. In this thesis, we deal with the following problems: • barycenters w.r.t. a general transportation cost, their existence and stability; • concentration and central limit theorem for empirical Wasserstein barycenters of Gaussian measures; • characterization, properties, and central limit theorem for entropy-penalized Wasserstein barycenters; • optimal transportation problem, regularized with the Dirichlet energy of a transport plan. Another part of the thesis is devoted to the complexity analysis of the iterative Bregman projections algorithm. This is a generalization of the well-known Sinkhorn algorithm, which allows us to find an approximate solution of the optimal transportation problem and the Wasserstein barycenter problem as well.