In this paper we present the developments conducted at LNE-LNHB with the aim of improving the knowledge of the shapes of beta spectra, concerning the theoretical calculation of the spectra as well as their experimental determination by means of cryogenic detectors. The theoretical calculation comprises a screening correction that avoids the non-physical discontinuity present in the standard correction, as well as a correction of the exchange effect that is usually not taken into account and has a strong impact on the spectra of 63Ni and 241Pu. These calculations are compared with experimental spectra obtained with metallic magnetic calorimeters, a class of cryogenic detectors. The beta emitter is enclosed in the absorber of these detectors which offer high detection efficiency, high energy resolution and a very low energy detection threshold. The measured spectra of 63Ni and 241Pu confirm the correctness of the theoretical spectra, in particular the calculation of the exchange effect down to very low energies that have been explored for the first time.