Outer approximations of continuous, discrete-time systems, or maps, incorporate bounded error and allow for the rigorous extraction of dynamics via Conley Index theory and other tools. Recent advances, including joint work with W. Kalies, M. D. LaMar, R. Trevi'no and R. Frongillo, have extended the class of maps to which these methods may be applied and improved the types of results that one may obtain, specifically in terms of computed lower bounds on topological entropy. I will describe some of this recent work and show sample results for systems ranging from the two-dimensional Henon map to the infinite-dimensional Kot-Schaffer integrodifference operator from ecology.