In this talk, I will discuss my recent work on entropic optimal transport (EOT). In the first part, I will discuss limit theorems for EOT maps, dual potentials, and the Sinkhorn divergence. The key technical tool we use is a first and second-order Hadamard differentiability analysis of EOT potentials with respect to the marginal distributions, from which the limit theorems, bootstrap consistency, and asymptotic efficiency of the empirical estimators follow. The second part concerns the entropic Gromov-Wasserstein (EGW) distance, which serves as a computationally efficient proxy for the Gromov-Wasserstein distance. By leveraging a variational representation that ties the EGW problem with a series of EOT problems, we derive stability results of EGW with respect to the auxiliary matrix, which enables us to develop efficient algorithms for solving the EGW problem. This talk is based on joint work with Ziv Goldfeld, Gabriel Rioux, and Ritwik Sadhu.