A general theory of the two-state curve-crossing problem has been developed, with a complete solution of an accurate model for "close" crossings (including numerical computations for strong coupling). Results clarify the position of the Landau-Zener approximation and its improvements by Nikitin and others, provide a general way of extending these approximations into regions often treated incorrectly (including the high-energy limit), and can be readily adapted to simple, rapid calculations.