The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. The author defines a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. She shows the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in fixed point theory. Table of Contents  A review of fixed point theory
 The converse to the Lefschetz fixed point theorem
 Topological duality and fixed point theory
 Why bicategories?
 Duality for parametrized modules
 Classical fixed point theory
 Duality for fiberwise parametrized modules
 Fiberwise fixed point theory
 A review of bicategory theory
 Index
 Index of notation
 Bibliography
