By the twisted trace formula for the Hecke correspondences we understand the trace formulas for Hecke operators for a non-connected algebraic group acting on certain cohomology spaces. The traces of Hecke operators developed by Selberg in 1957 and the work of Arthur on the traces of Hecke operators of 1989 on L 2 -cohomology are in some sense application of analysis and representation theory. On the other hand the work of Eichler in 1957 and of Goresky-MacPherson, Harder and Kottwitz from 1993

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... o present, reflects more geometric and topological applications. These may be seen as different solutions to the same problem, the calculation of the traces of Hecke operators. The present paper intends to be an introductory note about these operators and their applications.