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We report on a study of the evolution of magnetic fields of neutron stars,driven by the expulsion of magnetic flux out of the proton superconducting coreof the star. The rate of expulsion, or equivalently the velocity of outwardmotion of flux-carrying proton-vortices is determined from a solution of theirequation of motion. A determination of the effective forces on the fluxoidsmoving through the quantum liquid interior of neutron stars is howeverconfronted with many ambiguities about the properties of this special case ofsuperconductivity in the nature. Also, the behaviour of the fluxoids at thecore boundary, and the subsequent evolution of the expelled flux within thehighly conductive surrounding crust, are other related issues that have notbeen so far explored in any great details.Comment: 8 papegs, 2 figures, accepted at the 1st Regional Conference on  Magnetic and Superconducting Materials (MSM-99), Tehran, Sept. 199

SUPERCONDUCTIVITY AND MAGNETISM AT NUCLEAR-MATTER DENSITIES: AN ASTRONOMICAL CHALLENGE