This is a lecture note series on mathematical topics, edited by A. Dold and B. Eckmann. The series includes notes from a seminar on the work of A. Grothendieck, held at Harvard in 1963/64. The notes are based on a seminar that was suggested by Robin Hartshorne, who was a junior fellow at Harvard. The seminar was focused on Grothendieck's theory of duality for coherent sheaves, which had been hinted at in his talks to the Séminaire Bourbaki in 1957 and to the International Congress of Mathematicians in 1958, but had not been systematically developed. Grothendieck agreed to provide an outline of the material, while Hartshorne would fill in the details and write up the lecture notes. The seminar took place in the fall and winter of 1963-64, with the assistance of several mathematicians, including David Mumford, John Tate, Stephen Lichtenbaum, and John Fogarty. The seminar resulted in six exposés, which were circulated to a limited audience under the title "Séminaire Hartshorne." The present notes are a revised, expanded, and completed version of the previous notes. The notes cover topics such as the derived category, applications to preschemes, duality for projective morphisms, local cohomology, dualizing complexes and local duality, residual complexes, and the duality theorem. The book also includes an index of definitions, an index of notations, a bibliography, and an appendix by P. Deligne on cohomology with support and the construction of the functor f!. The notes are intended to provide a comprehensive overview of Grothendieck's theory of duality for coherent sheaves.This is a lecture note series on mathematical topics, edited by A. Dold and B. Eckmann. The series includes notes from a seminar on the work of A. Grothendieck, held at Harvard in 1963/64. The notes are based on a seminar that was suggested by Robin Hartshorne, who was a junior fellow at Harvard. The seminar was focused on Grothendieck's theory of duality for coherent sheaves, which had been hinted at in his talks to the Séminaire Bourbaki in 1957 and to the International Congress of Mathematicians in 1958, but had not been systematically developed. Grothendieck agreed to provide an outline of the material, while Hartshorne would fill in the details and write up the lecture notes. The seminar took place in the fall and winter of 1963-64, with the assistance of several mathematicians, including David Mumford, John Tate, Stephen Lichtenbaum, and John Fogarty. The seminar resulted in six exposés, which were circulated to a limited audience under the title "Séminaire Hartshorne." The present notes are a revised, expanded, and completed version of the previous notes. The notes cover topics such as the derived category, applications to preschemes, duality for projective morphisms, local cohomology, dualizing complexes and local duality, residual complexes, and the duality theorem. The book also includes an index of definitions, an index of notations, a bibliography, and an appendix by P. Deligne on cohomology with support and the construction of the functor f!. The notes are intended to provide a comprehensive overview of Grothendieck's theory of duality for coherent sheaves.