  The radiative decays of the pseudoscalar mesons $P=(\pi^0,\eta,\eta')$ are of
special interest as they provide an experimental window on the electromagnetic
and colour chiral $U_A(1)$ anomalies. $\pi^0\to\gamma\gamma$ is well described
by the electromagnetic $U_A(1)$ anomaly under the assumption that $\pi^0$ is a
Goldstone boson for spontaneously broken chiral symmetry, but the analogous
results for $\eta'(\eta)\to\gamma\gamma$ are complicated by the colour $U_A(1)$
anomaly in QCD. This paper reviews the theory of $\eta'(\eta)\to\gamma\gamma$
decays, emphasising the role of the colour $U_A(1)$ anomaly and the
renormalisation group. The relation to the Witten-Veneziano mass formula for
the $\eta'$ and the QCD topological susceptibility is derived. The implications
for the phenomenological analysis of $P\to\gamma\gamma$ are described and a
proposal is made for a comprehensive re-analysis of existing data on
$P\to\gamma\gamma$, $P\to V\gamma$, where $V=(\rho,\omega, \phi)$ is a light
$1^-$ meson, $\eta'(\eta)\to \pi\pi\gamma$, $\psi\to\eta'(\eta)\gamma$, etc.
Other applications, to the muon anomalous magnetic moment $g_\m-2$ and the
determination of the photon structure function $g_1^\gamma$ in polarised deep
inelastic scattering, are also briefly discussed.

High Energy Physics - Phenomenology