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We derive a Melvin Universe type solution describing a magnetic fieldpermeating the whole Universe in gravity minimally coupled to any non-linearelectromagnetic theory, including Born-Infeld Theory. For a large set ofnon-linear electrodynamics theories, our solution is complete and non-singular,as long as the magnetic field is sub-critical. We examine some properties ofthe solution; in particular there is a shift of the symmetry axis and anon-standard period along the orbits of the U(1) symmetry to avoid a conicalsingularity. We show these are consistent with the usual Dirac quantizationcondition for the magnetic flux. We find exact solutions describing propagationof waves in the `generalized Melvin Universe' along the principal nulldirections of the electromagnetic field, where the Boillat and Einsteinlight-cone touch. By electric-magnetic duality we show that similar Melvinelectric and dyonic universes can be obtained.Comment: LaTeX, 15 page

The Melvin universe in Born-Infeld theory and other theories of nonlinear electrodynamics