In this thesis we attempt to generalize some of Kummer's work on Fermat's Last Theorem over the rational numbers to quadratic fields. In particular, under certain congruence conditions it is shown that the Fermat equation of exponent p has no solution over Q(√m) when p is a m-regular prime. Completely analogous to the work of Kummer, it is shown that m-regular primes can be described in terms of the generalized Bernoulli numbers. When p = 3,5 and 7, an explicit, easily computable criterion is given for m-regularity.