We present a nonperturbative lattice calculation of the form factors which contribute to the amplitudes for the radiative decays P -> l (v) over bar (l gamma), where P is a pseudoscalar meson and l is a charged lepton. Together with the nonperturbative determination of the corrections to the processes P -> l (v) over bar (l),= due to the exchange of a virtual photon, this allows accurate predictions at O(alpha(em)) to be made for leptonic decay rates for pseudoscalar mesons ranging from the pion to the D-s meson. We are able to separate unambiguously and nonpertubatively the pointlike contribution, from the structure-dependent, infrared-safe, terms in the amplitude. The fully nonperturbative O(alpha) improved calculation of the inclusive leptonic decay rates will lead to the determination of the corresponding Cabibbo-Kobayashi-Maskawa matrix elements also at O(alpha(em)). Prospects for a precise evaluation of leptonic decay rates with emission of a hard photon are also very interesting, especially for the decays of heavy D and B mesons for which currently only modeldependent predictions are available to compare with existing experimental data.