A fixed point of a gravitational lensing map represents those positions from which a pointlike light source has a lensed image that, despite gravitational lensing, corresponds to the original position of the source. In this paper we study fixed points of lensing maps due to a generic gravitational lens with applications to nonsingular isolated lenses, and to point-mass lenses with continuous matter and shear. Counting formulas and bounds on the number of fixed points are determined. The results include an odd-number fixed-point theorem for nonsingular isolated deflectors. Information on the positions of fixed points are found for the case of point masses on a lens plane with and without shear. The methods of the paper are based on Morse theory, complex variables, and resultants. © 1998 American Institute of Physics.