The decay of 70-min Te129 and 33-day Te129m has been studied using a beta-ray spectrometer, scintillation spectrometers, and coincidence techniques. The 70-min activity, studied in equilibrium with Te129m and also independently, was seen to emit gamma rays of energies 27, 205, 245, 275, 350, 455, 550, 625, 755, 810, 825, 960, 1080, and 1235 keV. From coincidence relationships among these gamma rays, it was concluded that levels at 27, 275, 350 or 755, 482, 550, 810, 1105, and 1235 keV are excited in the decay of the 70-min activity. Te129m was observed to decay by the emission of gamma rays of energies 550, 695, 720, and 830 keV in addition to the 106-keV isomeric transition. These gamma rays were attributed to the excitation of levels at 695, 830, 1385, and 1415 keV. The Fermi plot of the beta spectrum of Te129m studied with a Siegbahn-Slätis spectrometer showed beta groups with end-point energies of 1595, 1452, 976, and 690 keV. From beta-gamma coincidence measurements it was concluded that 1595 keV is the end-point energy of the beta transition from Te129m to the ground state of I129 while 1452 keV is the end-point energy of the beta transition from the 70-min activity to the 27-keV level. Comparison of the total intensities of the beta transitions from Te129m and Te129 showed that the isomeric transition takes place with a branching ratio of 0.64. The ft values for the various beta groups were found from their relative intensities and spin assignments to various levels were made on this basis.