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This question is inspired by, but is independent of: http://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles

Line bundles are classified by $H^1(X,\mathcal{O}^\times_X)$. We also know that in general that $H^1(X,G)$, where $G$ is a sheaf from open sets in $X$ to $Grps$, classifies $G$-torsors over X.