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This article is based on lecture notes prepared for the August 2006 CologneSummer School. The first part contains background material and references forbeginners. The second (and main) part is a survey of the current status in thetheory of pluri-theta linear series and generalized theta divisors on modulispaces of vector bundles on curves. It emphasizes relatively new techniquesemployed in the analysis of linear series on these moduli spaces, namely theuse of moduli spaces of stable maps for understanding Quot schemes, and theFourier-Mukai functor in the study of coherent sheaves on abelian varieties. Inaddition, it briefly describes recent important developments, most significantof which is the proof of the Strange Duality conjecture due to Belkale andMarian-Oprea.

Generalized theta linear series on moduli spaces of vector bundles on curves