Associated with an electromagnetic disturbance is a mass, the gravitational attraction of which under appropriate circumstances is capable of holding the disturbance together for a time long in comparison with the characteristic periods of the system. Such gravitational-electromagnetic entities, or "geons" are analyzed via classical relativity theory. They furnish for the first time a completely classical, divergence-free, self-consistent picture of the Newtonian concept of body over the range of masses from ~1039 g to ~1057 g. Smaller geons are quantum objects whose analysis would call for the treatment of characteristic new effects. Topics covered in the discussion include: 1. Need for a self-consistent formulation of the concept of "body" in classical physics; geons vs free waves; electrical neutrality of geon; size and mass relations; the quantum limit and electron pair phenomena. 2. Orders of magnitude for toroidal geons; first estimates of leakage rates; a "phosphor" model of a geon; attrition and attritivity; energy action relation. 3. Idealized spherical geon; conditions required for symmetry; instability relative to pairing of light rays; time scale of instability long compared to vibration periods; spherical metric; wave equation for electromagnetic potential; evaluation of stress-energy tensor; its position as source of gravitation field; the gravitational field equations; the three equations of the self-consistent geon; simplification by scale transformation; first analysis of the eigenvalue problem; further scale transformation to get behavior of solution in active region of geon; further analysis of eigenvalue dependence; electronic calculator integration of equations of self-consistent geon; mass and radius values. 4. Transformations and interactions of electromagnetic geons; evaluation of refractive index barrier penetration integral for spherical geon; photon-photon collision processes as additional mechanism for escape of energy from system; restatement in language of coupling of characteristic modes; the thermal geon; comparison of gravitation and virtual electron pair phenomena as sources of coupling between modes; gravitational coupling and collective vibrations of geon; fission of a geon; interaction between two geons simple at large distances; orientation dependence and exponential term at intermediate distances; violent transmutation processes in closer encounters. 5. Influence of virtual pairs on geon structure; description in terms of refractive index correction; relation to photon-photon collision picture; more precise formulation via Heisenberg-Euler electrodynamics; corrections to stress-energy tensor and electromagnetic field equations. 6. Neutrino-containing geons; general similarity to electromagnetic geons; specificity of geon-geon interactions; the size subject to simple analysis unexpectedly limited by neutrino-neutrino encounters and the process ν+ν-->μ+e similarity of size limitation to that for electromagnetic geons; comments on present status of neutrino theory of light. 7. Electricity, Gauss's theorem, and gravitational field fluctuations. 8. Conclusions: the geon completes the scheme of classical physics; one's interest in following geons into quantum domain will depend upon one's view of the relation between very small geons and elementary particles.