Abstract: Precise data on radiative leptonic τ decays offer the opportunity to probe the electromagnetic properties of the τ and may allow to determine its anomalous magnetic moment which, in spite of its precise Standard Model prediction, has never been measured. Recently, the branching fractions of the radiative leptonic τ decays (τ→lνγ, with l=e,μ) were measured by the Babar Collaboration. These precise measurements, with a relative error of about 3%, must be compared with the branching ratios at the next-to-leading order in QED. Indeed the radiative corrections are expected to be of order 10%, for l=e, and 3%, for l=μ. Here, we present the prediction of the differential decay rates and branching ratios of the radiative μ and τ leptonic decays in the Standard Model at the next-to-leading order and we compare them with the recent Babar measurements. Moreover, we report on a dedicated feasibility study for the measurements of the τ anomalous magnetic moment at Belle and Belle II.