Newly observed decay schemes of the nuclei 137Sb and 138Sb are reported. The neutron-rich Sb isotopes were produced by the in-flight fragmentation of a 238U primary beam with an energy of 345 MeV/nucleon. Several new excited states of 137Te with tentatively assigned spin-parities of (5/2-), (9/2-), and (7/2) have been established which play an important role in the evolution of neutron levels beyond N=82. The study of the ß decay of 138Sb led to a considerable extension of the level scheme of 138Te including the identification of several nonyrast states. The structure of 137Te and 138Te is discussed on the basis of large-scale shell-model calculations performed using two different effective interactions.