experts in the ?eld. Building on lectures by one of us (J. R.) at the CERN Accelerator School, and motivated by the positive response to a series of seminars on ¿FEL t- ory for pedestrians¿, given by P. S. within the framework of the Academic Training Program at DESY, we have aimed at presenting the theory of the low- gainandthehigh-gainFELinaclearandconcisemathematicallanguage. Particular emphasis is put on explaining and justifying the assumptions and approximations that are needed to obtain the di?erential equations descr- ing the FEL dynamics. Although we have tried our best to be ¿simple¿, the mathematical derivations are certainly not always as simple as one would like them to be. However, we are not aware of any easier approach to the FEL theory. Some of the more involved calculations are put into the appendices.