We evaluate the matrix elements for the radiative kaon decays K + → l + ν l γ, l + ν l l ′+ l ′− and K → πlν l γ (l, l ′ = e, µ) to next-to-leading order in chiral perturbation theory. We calculate total rates and rates with several kinematical cuts and confront the results with existing data. Measurements at future kaon facilities will allow for a more detailed comparison between theory and experiment. † Address after 1 November 1992: NORDITA,