Time reversibility plays an important role in disciplinesconcerning dynamical systems, e.g. in physics, statistical mechanics,stochastic processes, and biology. However, its use inproviding information-theoretic fundamental limits appears tobe somewhat limited. Recent developments at the intersectionof information theory and control have demonstrated howdirected information characterizes fundamentallimitations of stochastic systems with dynamics.In this talk, we connect time reversibility of Markov chainsto maximization of directed information for a classof stochastic dynamical systems. With this framework, wecharacterize the capacity of a class channels with (infinite) memoryand provide optimality "matching " conditions for sequential sequentialestimation of a random process over a communication channel withfeedback. Examples include communication over queuing timing channelsas well as Blackwell 's trapdoor (chemical) channel, along withsequential estimation within the context of decentralized control.