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MOSFiT (Modular Open-Source Fitter for Transients) downloads transient datasets from open online catalogs (e.g., the Open Supernova Catalog), generates Monte Carlo ensembles of semi-analytical light curve fits to those datasets and their associated Bayesian parameter posteriors, and optionally delivers the fitting results back to those same catalogs to make them available to the rest of the community. MOSFiT helps bridge the gap between observations and theory in time-domain astronomy; in addition to making the application of existing models and creation of new models as simple as possible, MOSFiT yields statistically robust predictions for transient characteristics, with a standard output format that includes all the setup information necessary to reproduce a given result.