  We present a comparison of existing experimental data for the radiative
leptonic decays $P\to\ell\nu_\ell\gamma$, where $P=K$ or $\pi$ and $\ell=e$ or
$\mu$, from the KLOE, PIBETA, E787, ISTRA+ and OKA collaborations with
theoretical predictions based on the recent non-perturbative determinations of
the structure-dependent vector and axial-vector form factors, $F_V$ and $F_A$
respectively. These were obtained using lattice QCD+QED simulations at order
$O(\alpha_{\mathrm{em}})$ in the electromagnetic coupling. We find good
agreement with the KLOE data on $K\to e\nu_e\gamma$ decays from which the form
factor $F^+=F_V+F_A$ can be determined. For $K\to\mu\nu_\mu\gamma$ decays we
observe differences of up to 3\,-\,4 standard deviations at large photon
energies between the theoretical predictions and the data from the E787, ISTRA+
and OKA experiments and similar discrepancies in some kinematical regions with
the PIBETA experiment on radiative pion decays. A global study of all the
kaon-decay data within the Standard Model results in a poor fit, largely
because at large photon energies the KLOE and E787 data cannot be reproduced
simultaneously in terms of the same form factor $F^+$. The discrepancy between
the theoretical and experimental values of the form factor $F^-=F_V-F_A$ is
even more pronounced. These observations motivate future improvements of both
the theoretical and experimental determinations of the structure-dependent form
factors $F^+$ and $F^-$, as well as further theoretical investigations of
models of "new physics" which might for example, include possible flavor
changing interactions beyond $V - A$ and/or non-universal corrections to the
lepton couplings.
