Fulfilling the rich promise of rapid advances in time-domain astronomy is only possible through confronting our observations with physical models and extracting the parameters that best describe what we see. Here, we introduce REDBACK; a Bayesian inference software package for electromagnetic transients. REDBACK provides an object-orientated PYTHON interface to over 12 different samplers and over 100 different models for kilonovae, supernovae, gamma-ray burst afterglows, tidal disruption events, engine-driven transients among other explosive transients. The models range in complexity from simple analytical and semi-analytical models to surrogates built upon numerical simulations accelerated via machine learning. REDBACK also provides a simple interface for downloading and processing data from various catalogues such as Swift and FINK. The software can also serve as an engine to simulate transients for telescopes such as the Zwicky Transient Facility and Vera Rubin with realistic cadences, limiting magnitudes, and sky coverage or a hypothetical user-constructed survey or a generic transient for target-of-opportunity observations with different telescopes. As a demonstration of its capabilities, we show how REDBACK can be used to jointly fit the spectrum and photometry of a kilonova, enabling a more powerful, holistic probe into the properties of a transient. We also showcase general examples of how REDBACK can be used as a tool to simulate transients for realistic surveys, fit models to real, simulated, or private data, multimessenger inference with gravitational waves, and serve as an end-to-end software toolkit for parameter estimation and interpreting the nature of electromagnetic transients.