The Serre-Swan theorem states (at least in one form) that the category of real vector bundles over a compact Hausdorff space $M$ is equivalent to the category of finitely generated projective modules over the ring $C(M)$ of continuous functions on $M$. The equivalence is provided by the functor $\Gamma$ sending a bundle to the totality of all its continuous sections.
Are there any classic applications or uses of this theorem? To me right now it seems like a pristine result to be admired from a distance, as I currently know of no actual use for it. I'd love to remedy that!