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The superheated Meissner state in type-I superconductors is studied both analytically and numerically within the framework of Ginzburg-Landau theory. Using the method of matched asymptotic expansions we have developed a systematic expansion for the solutions of the Ginzburg-Landau equations in the limit of small κ, and have determined the maximum superheating field H sh for the existence of the metastable, superheated Meissner state as an expansion in powers of κ 1/2. Our numerical solutions of these equations agree quite well with the asymptotic solutions for κ< 0.5. The same asymptotic methods are also used to study the stability of the solutions, as well as a modified version of the Ginzburg-Landau equations which incorporates nonlocal electrodynamics. Finally, we compare our numerical results for the superheating field for large κ against recent asymptotic results for large κ, and again find a close …