I will recall the notion of continuous trace C*-algebra, and present classical classification results for such C*-algebras and their automorphisms in terms of the Cech cohomology of the spectrum due to Dixmier--Douady and Phillips--Raeburn. I will then explain how the framework of Borel-definable homological algebra recently developed with Jeffrey Bergfalk and Aristotelis Panagiotopoulos allows one to refine the analysis and obtain more precise classification results.