The geometry of moduli spaces and stacks of vector bundles on curves have been intensively studied from different perspectives; for example, via point counting over finite fields by Harder and Narasimhan, and gauge theoretically by Atiyah and Bott over the complex numbers. Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will present a formula for the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah--Bott and Behrend--Dhillon. The proof involves rigidifying this stack using Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the cohomology of small maps, which is closely related to the Grothendieck-Springer resolution. I will explain how to extend this to a formula for the stack of coherent sheaves and, if there is time, I will give an overview of other motivic descriptions of closely related moduli spaces. This is joint work with Simon Pepin Lehalleur.

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