  Different approaches to describe Compton scattering and the polarizability of
the nucleon have been discussed up to now. We show that the most appropriate
ones are provided by non-subtracted dispersion theories of the fixed-$t$ and
fixed-$\theta$ types, where the properties of these two versions are
complementary so that advantage can be taken from both of them. In the frame of
fixed-$t$ dispersion theory it was possible to precisely reproduce experimental
differential cross sections obtained for the proton in a wide angular range and
for energies up to 1 GeV. At energies of the first resonance region and below,
precise values for the electromagnetic polarizabilities and
spin-polarizabilities have been determined for the proton and the neutron. The
data show that diamagnetism is a prominent property of nucleon structure, where
the underlying mechanism is a $t$-channel $\sigma$-meson exchange. A similar
mechanism is responsible for the backward spin-polarizability where the
relevant meson is the $\pi^0$. It is a challenge for further research to
integrate the $\sigma$ and $\pi^0$ intermediate states into a consistent
description of the structure of the nucleon.
