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We present a rigorous analysis of the Bardeen-Cooper-Schrieffer (BCS) model for general pair interaction potentials \lambda V. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain effective linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. We give necessary and sufficient conditions for the existence of a positive critical temperature. Moreover we present a precise calculation of the asymtotic behavior of the critical temperature in the limit \lambda to 0, which is {\em not} governed by the scattering length.