The β decay of the neutron-rich 134In and 135In was investigated experimentally in order to provide new insights into the nuclear structure of the tin isotopes with magic proton number Z = 50 above the N = 82 shell. The β-delayed γ -ray spectroscopy measurement was performed at the ISOLDE facility at CERN, where indium isotopes were selectively laser-ionized and on-line mass separated. Three β-decay branches of 134In were established, two of which were observed for the first time. Population of neutron-unbound states decaying via γ rays was identified in the two daughter nuclei of 134In, 134Sn and 133Sn, at excitation energies exceeding the neutron separation energy by 1 MeV. The β-delayed one- and two-neutron emission branching ratios of 134In were determined and compared with theoretical calculations. The β-delayed one-neutron decay was observed to be dominant β-decay branch of 134In even though the Gamow-Teller resonance is located substantially above the two-neutron separation energy of 134Sn. Transitions following the β decay of 135In are reported for the first time, including γ rays tentatively attributed to 135Sn. In total, six new levels were identified in 134Sn on the basis of the βγγ coincidences observed in the 134In and 135In β decays. A transition that might be a candidate for deexciting the missing neutron single-particle 13/2+ state in 133Sn was observed in both β decays and its assignment is discussed. Experimental level schemes of 134Sn and 135Sn are compared with shell-model predictions. Using the fast timing technique, half-lives of the 2+, 4+, and 6+ levels in 134Sn were determined. From the lifetime of the 4+ state measured for the first time, an unexpectedly large B(E2; 4+ → 2+) transition strength was deduced, which is not reproduced by the shell-model calculations.