In the fall of 1963, there was a seminar at Harvard on duality theorems in algebraic geometry. Its aim was to understand how the systematic use of the revolutionary new method of derived categories had enabled Grothendieck to generalize to the relative case, for a proper morphism of schemes, the duality theorems of Serre for projective varieties over a field. What we present here is the type-written “prenotes” which he prepared to guide Hartshorne in running the seminar. Unpublished until now, they show the mathematician as he reached the height of his creative powers. This edition also contains a preface by Hartshorne explaining the origins of the ideas presented in the seminar, together with a discussion of the differences between these prenotes and his report in the Springer Lecture Notes Residues and Duality of 1966, where the derived category and these new duality theorems appeared for the first time in print.