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.

34 pages, 11 figures. v2: version published in JHEP, added substantial section on B and D meson electroweak decays, added comment on psi' to eta(')/eta_c gamma, Figs 5 and 6 split and made clearer, added references, other minor revisions which don't change conclusions