Many classical constructions in representation theory can be extended to vector bundles on algebraic curves. The classical Borel-Weil theorem on the geometric description of irreducible representations of simple algebraic groups has been extended to this setting by Teleman. I will introduce algebraic vector bundles and Borel-Weil-Bott-Teleman theory. Then I will describe its consequences on the derived category of the moduli space of vector bundles on a curve and the Fano visitor problem of Jacobian varieties. This is ongoing joint work with Kyoung-Seog Lee.