[en] Interest in the Atiyah-Singer index theorem and the closely related Atiyah-Bott-Lefshetz formula has increased sharply in recent years. This is explained by the discovery of the connection of the index theorem and Lefshetz type formulas with supersymmetric quantum theories. In this work, a brief survey of the directions in which the Atiyah-Bott theorem has been generalized are given. Elliptic complexes on a manifold with boundary whose differentials are Boutet de Monvel operators are studied. An Atiyah-Bott-Lefschetz type formula is obtained for such complexes. 50 refs., 1 fig