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LIGO and Virgo have now reported more than 50 gravitational-wave signals from coalescing compact binaries. The paths that these objects trace around each other in the fragments of seconds before they merge can contain a wealth of information about the previous lives of these objects. Binary compact objects have two overarching formation channels: isolated, in which the pair live from birth to death as a binary without any external influence, and dynamical, in which two compact objects become bound in a populous environment like a star cluster. Isolated binaries are expected to trace circular paths around each other close to merger. However, when binaries form dynamically, their paths can be elliptical. In this talk, I explain how different formation channels lead to different distributions of eccentricity. I present eccentricity measurements obtained for LIGO and Virgo's binary black holes, including new results from GWTC-2. Finally, I show how the orbital eccentricity of binaries in GWTC-2 have implications for every binary so far detected by LIGO and Virgo.