We derive and discuss the finite-energy sum rules, which form consistency conditions imposed by analyticity on the Regge analysis of a scattering amplitude. Their finite form makes them particularly useful in practical applications. We discuss the various applications, emphasizing a new kind of bootstrap predicting the Regge parameters from low-energy data alone. We apply our methods to πN charge exchange and are able to derive many interesting features of the high-energy amplitudes at various t. In particular, we establish the existence of zeros of the amplitudes and of additional ρ poles. On the basis of the finiteenergy sum rules and the analysis of the πN amplitudes, we present theoretical and experimental evidence that double counting is involved in the interference model, which adds direct-channel resonances to the exchanged Regge terms.