See Announcing a seminar for a little context.
The past four weeks I've given lectures on Grothendieck duality for the ANAGRAMS seminar (yes, this is a redundancy). The goal was to tell people something about Riemann-Roch and Serre duality first, then go on to Grothendieck duality and discuss some of the proofs. I changed the schedule a little while I was giving the lectures (originally I planned more on Grothendieck duality, but the down-to-earth geometric applications were more appropriate for the audience, I think). I have prepared notes for each of these lectures, which are now available at the seminar webpage.
Maybe I will rewrite some things for my weblog, if people want me to elaborate on certain aspects.
And in the philosophy of open research, there is the Git repository for the notes. Feel free to contribute!