The ensemble of chemical element abundance measurements for stars, along with precision distances and orbit properties, provides high-dimensional data to study the evolution of the Milky Way. With this third data release of the Galactic Archaeology with HERMES (GALAH) survey, we publish 678\,423 spectra for 588\,571 mostly nearby stars (81.2\% of stars are within $<$ 2 kpc), observed with the HERMES spectrograph at the Anglo-Australian Telescope.