I will discuss some results classifying holomorphic vector bundles on certain algebraic surfaces in terms of linear algebra data associated to a quiver, and extensions of this result to some threefolds. The surface results go back to work of Atiyah and others on instantons, extended by Kronheimer and Nakajima. The threefold results are much more recent; they connect nicely to some recent physics work on supersymmetric quiver gauge theory.