Low rank approximation methods are a central pillar of modern scientific computing. They are the powerhouse behind many fast and superfast methods that are relied upon on for computing solutions to various partial differential equations, linear systems, and matrix equations. In this talk, we focus on the role that rational approximation methods can play in the design of such algorithms. We illustrate how rational approximation tools can help us design highly effective low rank methods in the context of two very different (but surprisingly related!) kinds of problems: (1) the development of fast solvers for matrix equations and special linear systems, and (2) the development of solvers for PDEs involving functions of operators on geometrically complicated domains.