Optimal transport (OT) theory sharesconnections with geometry, analysis, probability theory, and otherfields in mathematics. The recent resurgence of interests in OT stems from appliedfields such as machine learning, image processing and statisticsthrough the introduction of entropic regularization. In this talk, we will discuss the convergence of entropically regularized optimaltransport. Our first result is about a large deviation principle ofthe associated optimizers in entropic OT and the second result is about the stability of the optimizers under weak convergence. To prove these results, we will introduce a new notion called 'cyclical invariance'of measures. This is a joint work with Marcel Nutz and Espen Bernton. If time permits, we touch on an ongoing work on the rate of convergence of the entropic OT.