  We derive a set of sum rules for the light-by-light scattering and fusion:
$\gamma\gamma \to all$, and verify them in lowest order QED calculations. A
prominent implication of these sum rules is the superconvergence of the
helicity-difference total cross-section for photon fusion, which in the hadron
sector reveals an intricate cancellation between the pseudoscalar and tensor
mesons. An experimental verification of superconvergence of the polarized
photon fusion into hadrons is called for, but will only be possible at $e^+
e^-$ and $\gamma\gamma$ colliders with both beams polarized. We also show how
the sum rules can be used to measure various contributions to the low-energy
light-by-light scattering.
