Twelve years after Atiyah's article classifying vector bundles on curves of genus one, Narasimhan and Ramanan published a lovely paper in the Annals, (89) no.2, 1969, p.14, where they solved the case of semi stable rank 2 vector bundles on genus two curves. This case is perhaps more typical of the higher genus situation. Basically, a rank two vector bundle is analyzed by producing a sub line bundle, whose quotient is also a line bundle, and then studying how the vector bundle is reconstructed as a twisted sum of those two line bundles. I cannot improve on the wonderful references given by Georges Elencwajg above, but I have a short 4 or 5 page set of notes from a lecture given by Daniele Arcara, in a graduate class of mine, summarizing the status of moduli of rank 2 bundles on curves in 2001, if there is some way to send it to you or attach it here as a pdf file.

Twelve years after Atiyah's article classifying vector bundles on curves of genus one, Narasimhan and Ramanan published a lovely paper in the Annals, (89) no.2, 1969, p.14, where they solved the case of semi stable rank 2 vector bundles on genus two curves. This case is perhaps more typical of the higher genus situation. Basically, a rank two vector bundle is analyzed by producing a sub line bundle, whose quotient is also a line bundle, and then studying how the vector bundle is reconstructed as a twisted sum of those two line bundles. I cannot improve on the wonderful references given by Georges Elencwajg above, but I have a short 4 or 5 page set of notes from a lecture given by Daniele Arcara, in a graduate class of mine, summarizing the status of moduli of rank 2 bundles on curves in 2001, if there is some way to send it to you or attach it here as a pdf file....

I sent them to your email address at Princeton.