It is pointed out that the definition of the inhomogeneous Lorentz group as a symmetry group breaks down in the presence of gravitational fields even when the dynamical effects of gravitational forces are completely negligible. An attempt is made to rederive the Lorentz group as an "asymptotic symmetry group" which leaves invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. By analyzing recent work of Bondi and others on gravitational radiation it is shown that, with apparently reasonable boundary conditions, one obtains not the Lorentz group but a larger group. The name "generalized Bondi-Metzner group" ("GBM group") is suggested for this larger group. It is shown that the GBM group contains an Abelian normal subgroup whose factor group is isomorphic to the homogeneous orthochronous Lorentz group; that the GBM group contains precisely one Abelian four-dimensional normal subgroup, which can be identified with the group of rigid translations; that the GBM group contains an infinite number of different subgroups isomorphic to the inhomogeneous orthochronous Lorentz group; that the infinitesimal GBM group algebra permits at least one nontrivial representation, which is directly analogous to the rest-mass-zero and spin-zero representation of the Lorentz group; that in any representation of the infinitesimal GBM group algebra there is a "rest mass" operator which commutes with all the other operations; and that no similar "spin" operator appears to exist. It is argued that the GBM group is so similar to the inhomogeneous Lorentz group that the former may be compatible as a symmetry group with present microphysics. Two applications are given: Certain quantum commutation relations covariant under GBM transformations are presented; and a denumerably infinite set of integral invariants, for classical asymptotically flat gravitational fields, are derived. The four simplest integral invariants constitute the total energy momentum radiated to infinity by gravitational waves.