Wasserstein barycenters, introduced by M. Agueh and G. Carlier in 2011, is a non-linear average of probability measures with respect to the 2-Wasserstein geometry. Unfortunately, little is understood about their analytical and statistical properties outside of specific cases. This talk focuses on entropy-penalized Wasserstein barycenters, a regularization approach proposed by J. Bigot, E. Cazelles, and N. Papadakis in 2019. A key feature of these barycenters is their characterization through a fixed-point equation, which connects the density and the associated Kantorovich potentials. I will demonstrate how this characterization leads to important regularity properties, including moment bounds, smoothness, and maximum principle. Finally, we will discuss the statistical properties of entropic barycenters, covering the law of large numbers, the large deviation principle, and the functional central limit theorem in the regular setting. The talk is based on the joint work with G. Carlier and K. Eichinger.