Aims. We describe the design, construction, and characterization of the Band 9 heterodyne receivers (600-720 GHz) for the Atacama Large Millimeter/submillimeter Array (ALMA). First-light Band 9 data, obtained during ALMA commissioning and science verification phases, are presented as well. Methods. The ALMA Band 9 receiver units (so-called "cartridges"), which are installed in the telescope's front end, have been designed to detect and down-convert two orthogonal linear polarization components of the light collected by the ALMA antennas. The light entering the front end is refocused with a compact arrangement of mirrors, which is fully contained within the cartridge. The arrangement contains a grid to separate the polarizations and two beam splitters to combine each resulting beam with a local oscillator signal. The combined beams are fed into independent double-sideband mixers, each with a corrugated feedhorn coupling the radiation by way of a waveguide with backshort cavity into an impedance-tuned superconductor-insulator-superconductor (SIS) junction that performs the heterodyne down-conversion. Finally, the generated intermediate frequency (IF) signals are amplified by cryogenic and room-temperature HEMT amplifiers and exported to the telescope's IF back end for further processing and, finally, correlation. Results. The receivers have been constructed and tested in the laboratory and they show an excellent performance, complying with ALMA requirements. Performance statistics on all 73 Band 9 receivers are reported. Importantly, two different tunnel-barrier technologies (necessitating different tuning circuits) for the SIS junctions have been used, namely conventional AlO x barriers and the more recent high-current-density AlN barriers. On-sky characterization and tests of the performance of the Band 9 cartridges are presented using commissioning data. Continuum and line images of the low-mass protobinary IRAS 16293-2422 are presented which were obtained as part of the ALMA science verification program. An 8 GHz wide Band 9 spectrum extracted over a 0.3 × 0.3 region near source B, containing more than 100 emission lines, illustrates the quality of the data.