This volume contains the lectures of the Grothendieck seminar at the IHES in 1965–66, originally distributed as mimeographed notes. The only significant changes from the original version are that Exposé II is not included and Exposé III has been completely rewritten and expanded with an appendix III B. Other lectures have been left unchanged except for minor modifications and additions of footnotes. The core of the seminar is the Lefschetz formula in étale cohomology (III, III B, XII) and its application to the cohomological interpretation of L-functions (XIV). All results announced by Grothendieck in his Bourbaki talk are fully proved here. The trace formulas established in (III, III B, and XII), through different paths, are more general than needed to prove the rationality of L-functions. A shorter and more complete proof of this rationality, following Grothendieck's methods from XII and XIV, is given in Deligne's lecture (SGA 4 1/2 Rapport). We hope that the formulas in III, III B may be useful in other situations. The rest of the seminar consists of two lectures on the theory of limits leading to étale cohomology (V, VI), and various complements to the duality formalism (I, VII) and the Lefschetz formula (VII, X). Exposé I is independent of the rest of the seminar. Its main result is that, on a regular scheme satisfying certain local conditions, and under the assumption of resolution of singularities and purity, the constant sheaf of values Z/nZ (for n prime to the residual characteristic) is dualizing (I.3.4.1). This theorem applies to excellent regular schemes of characteristic zero, thanks to Hironaka and Artin (SGA 4 XIX). The case of a regular scheme of dimension 1 is treated separately by another method. Deligne proves in (SGA 4 1/2 Th. Finitude) that, if f: X → S is a finite type scheme over a regular scheme of dimension 0 or 1, the complex f^{Z/nZ}_c (n prime to the residual characteristic) is dualizing. However, it is unclear whether this result extends to regular excellent schemes of dimension > 1. Exposé II, which was titled "Künneth formulas for cohomology with arbitrary supports," is not included in this volume. It was written by L. Illusie, based on handwritten notes by Grothendieck, and concerned theorems of cohomological properness and local acyclicity, but these were only proved under resolution hypotheses. A proof of the same results without these hypotheses, obtained by Deligne (SGA 4 1/2 Th. Finitude), has made the publication ofThis volume contains the lectures of the Grothendieck seminar at the IHES in 1965–66, originally distributed as mimeographed notes. The only significant changes from the original version are that Exposé II is not included and Exposé III has been completely rewritten and expanded with an appendix III B. Other lectures have been left unchanged except for minor modifications and additions of footnotes. The core of the seminar is the Lefschetz formula in étale cohomology (III, III B, XII) and its application to the cohomological interpretation of L-functions (XIV). All results announced by Grothendieck in his Bourbaki talk are fully proved here. The trace formulas established in (III, III B, and XII), through different paths, are more general than needed to prove the rationality of L-functions. A shorter and more complete proof of this rationality, following Grothendieck's methods from XII and XIV, is given in Deligne's lecture (SGA 4 1/2 Rapport). We hope that the formulas in III, III B may be useful in other situations. The rest of the seminar consists of two lectures on the theory of limits leading to étale cohomology (V, VI), and various complements to the duality formalism (I, VII) and the Lefschetz formula (VII, X). Exposé I is independent of the rest of the seminar. Its main result is that, on a regular scheme satisfying certain local conditions, and under the assumption of resolution of singularities and purity, the constant sheaf of values Z/nZ (for n prime to the residual characteristic) is dualizing (I.3.4.1). This theorem applies to excellent regular schemes of characteristic zero, thanks to Hironaka and Artin (SGA 4 XIX). The case of a regular scheme of dimension 1 is treated separately by another method. Deligne proves in (SGA 4 1/2 Th. Finitude) that, if f: X → S is a finite type scheme over a regular scheme of dimension 0 or 1, the complex f^{Z/nZ}_c (n prime to the residual characteristic) is dualizing. However, it is unclear whether this result extends to regular excellent schemes of dimension > 1. Exposé II, which was titled "Künneth formulas for cohomology with arbitrary supports," is not included in this volume. It was written by L. Illusie, based on handwritten notes by Grothendieck, and concerned theorems of cohomological properness and local acyclicity, but these were only proved under resolution hypotheses. A proof of the same results without these hypotheses, obtained by Deligne (SGA 4 1/2 Th. Finitude), has made the publication of