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I'll start with some background about how circularly orderable groups arise in 3-manifold topology, and a conjecture of my friend seeking to characterize fibered hyperbolic 3-folds in this context. But for the most part, I'll just take all this as motivation to study circular orders on groups abstractly, and discuss various results and constructions we found in analogy with linear orders. An underlying theme is topological dynamics on S^1.