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A priori, there is nothing very special about shear-free or asymptotically shear-free nullgeodesic congruences. Surprisingly, however, they turn out to possess a large number offascinating geometric properties and to be closely related, in the context of general relativity,to a variety of physically significant effects. It is the purpose of this paper to try to fullydevelop these issues.This work starts with a detailed exposition of the theory of shear-free and asymptoticallyshear-free null geodesic congruences, i.e., congruences with shear that vanishes at future conformalnull infinity. A major portion of the exposition lies in the analysis of the space ofregular shear-free and asymptotically shear-free null geodesic congruences. This analysis leadsto the space of complex analytic curves in complex Minkowski space. They in turn play adominant role in the applications.The applications center around the problem of extracting interior physical properties of anasymptotically-flat spacetime directly from the asymptotic gravitational (and Maxwell) fielditself, in analogy with the determination of total charge by an integral over the Maxwell fieldat infinity or the identification of the interior mass (and its loss) by (Bondis) integrals of theWeyl tensor, also at infinity.More specifically, we will see that the asymptotically shear-free congruences lead us toan asymptotic definition of the center-of-mass and its equations of motion. This includes akinematic meaning, in terms of the center-of-mass motion, for the Bondi three-momentum. Inaddition, we obtain insights into intrinsic spin and, in general, angular momentum, includingan angular-momentumconservation law with well-defined flux terms. When a Maxwell fieldis present, the asymptotically shear-free congruences allow us to determine/define at infinitya center-of-charge world line and intrinsic magnetic dipole moment.