In recent years, the exploitation of the recoil-decay tagging (RDT) technique with large arrays of germanium detectors has revealed much information about the structure of heavy nuclei approaching the proton drip line. The yrast bands of the N ≤ 93 osmium isotopes have been identified in a campaign of tagging experiments using various spectrometer arrays coupled to the RITU gas-filled separator based at the University of Jyväskylä. Trends in the yrast state excitation energies have indicated a transition from γ-soft triaxial to near-spherical shapes with decreasing neutron number. Recent experimental results for 162Os obtained with the JUROGAM and GREAT spectrometers also indicate the importance of configurations involving the h9/2 neutron states as the N = Z = 82 shell gaps are approached.