The Binary Population and Spectral Synthesis suite of binary stellar evolution models and synthetic stellar populations provides a framework for the physically motivated analysis of both the integrated light from distant stellar populations and the detailed properties of those nearby. We present a new version 2.1 data release of these models, detailing the methodology by which Binary Population and Spectral Synthesis incorporates binary mass transfer and its effect on stellar evolution pathways, as well as the construction of simple stellar populations. We demonstrate key tests of the latest Binary Population and Spectral Synthesis model suite demonstrating its ability to reproduce the colours and derived properties of resolved stellar populations, including well-constrained eclipsing binaries. We consider observational constraints on the ratio of massive star types and the distribution of stellar remnant masses. We describe the identification of supernova progenitors in our models, and demonstrate a good agreement to the properties of observed progenitors. We also test our models against photometric and spectroscopic observations of unresolved stellar populations, both in the local and distant Universe, finding that binary models provide a self-consistent explanation for observed galaxy properties across a broad redshift range. Finally, we carefully describe the limitations of our models, and areas where we expect to see significant improvement in future versions.