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Given a group G, and a manifold M, can one describe all the ways that G acts on M? This is a remarkably rich question even in the case where M is the line or the circle, and is connected to problems in dynamics, topology, and foliation theory. This talk will describe one very useful way to capture such an action, namely, through the algebraic data of a left-invariant linear or circular order on a group. I'll explain new work, joint with C. Rivas, that relates the topology of the space of orders on a group G to the moduli space of actions of G on the line or circle. As an application we'll see new rigidity phenomena for actions, and the answers to some older questions about orderings.