  The Drell-Hearn-Gerasimov and Bjorken sum rules are special examples of
dispersive sum rules for the spin-dependent structure function G_1(\nu, Q^2) at
Q^2=0 and \infty. We generalize these sum rules through studying the
virtual-photon Compton amplitudes S_1(\nu, Q^2) and S_2(\nu,Q^2). At small Q^2,
we calculate the Compton amplitudes at leading-order in chiral perturbation
theory; the resulting sum rules can be tested by data soon available from
Jefferson Lab. For Q^2>>\Lambda_{QCD}^2, the standard twist-expansion for the
Compton amplitudes leads to the well-known deep-inelastic sum rules. Although
the situation is still relatively unclear in a small intermediate-Q^2 window,
we argue that chiral perturbation theory and the twist-expansion alone already
provide strong constraints on the Q^2-evolution of the G_1(\nu,Q^2) sum rule
from Q^2=0 to Q^2=\infty.
