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This paper reviews the progress made in the last few years in theoretical understanding of the properties of superconductors in very high magnetic fields. The key ingredient in the new understanding is the recognition that the usual negative effect of orbital frustration in a superconducting state, reflected in diamagnetic pair breaking, is obviated when all electrons reside in only the few lowest Landau levels. A new relation between the superconducting order parameter and the magnetic field now exists which permits a strong enhancement of the critical temperature with increasing magnetic field. The issue of Pauli pair breaking and the effect of impurities in this new state are also discussed, along with the important observation that the attractive component of the effective electron-electron interaction in systems with low carrier density is enhanced with increasing magnetic field. While this paper emphasizes the theoretical aspects of this high-field limit, it does present a unified picture of superconductivity throughout the whole temperature and magnetic-field phase diagram. Numerical applications to simple models of low-carrierdensity semiconductors and semimetals are also discussed, since these materials are the most likely candidates for this new phase. CONTENTS I. II.v. ! One final comment about diamagnetic pair breaking concerns the role of impurities. The addition of nonmagnetic impurities, not surprisingly, increases the "mass" of the Cooper pairs and makes a superconductor more type II. In fact (as we discuss in Sec. II), in the limit of high impurity density, the behavior of the H c2 ( T) line is correct all the way to T =0, and the crossover region to i/ c oo (Fig. 1) disappears.

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