Here we give a brief review on the current bounds on the general Majorana transition neutrino magnetic moments (TNMM) which cover also the conventional neutrino magnetic moments (NMM). Leptonic CP phases play a key role in constraining TNMMs. While the Borexino experiment is the most sensitive to the TNMM magnitudes, one needs complementary information from reactor and accelerator experiments in order to probe the complex CP phases.