  The composition of the eta and eta' mesons has long been a source of
discussion and is of current interest with new experimental results appearing.
We investigate what can be learnt from a number of different processes: V to P
gamma and P to V gamma (V and P are light vector and pseudoscalar mesons
respectively), P to gamma gamma, J/psi,psi' to P gamma, J/psi,psi' to P V, and
chi_{c0,2} to PP. These constrain the eta-eta' mixing angle to a consistent
value, phi approx 42 degrees; we find that the c cbar components are lesssim 5%
in amplitude. We also find that, while the data hint at a small gluonic
component in the eta', the conclusions depend sensitively on unknown form
factors associated with exclusive dynamics. In addition, we predict BR(psi' to
eta' gamma) approx 1 10^{-5} and BR(chi_{c0} to eta eta') approx 2 10^{-5} - 1
10^{-4}. We provide a method to test the mixing using chi_{c2} to eta eta, eta'
eta', and eta eta' modes and make some general observations on chi_{c0,2}
decays. We also survey the semileptonic and hadronic decays of bottom and
charmed mesons and find some modes where the mixing angle can be extracted
cleanly with the current experimental data, some where more data will allow
this, and some where a more detailed knowledge of the different amplitudes is
required.
