This paper is an overview of results that have been obtain in [J. Bigot, E. Cazelles, and N. Papadakis. Penalized barycenters in the Wasserstein space. Submitted. Available at https://128.84.21.199/abs/1606.010252] on the convex regularization of Wasserstein barycenters for random measures supported on Rd. We discuss the existence and uniqueness of such barycenters for a large class of regularizing functions. A stability result of regularized barycenters in terms of Bregman distance associated to the convex regularization term is also given. Additionally we discuss the convergence of the regularized empirical barycenter of a set of n iid random probability measures towards its population counterpart in the real line case, and we discuss its rate of convergence. This approach is shown to be appropriate for the statistical analysis of discrete or absolutely continuous random measures. In this setting, we propose an efficient minimization algorithm based on accelerated gradient descent for the computation of regularized Wasserstein barycenters.