We present the first major release of data from the SAMI Galaxy Survey. This data release focuses on the emission-line physics of galaxies. Data Release One includes data for 772 galaxies, about 20% of the full survey. Galaxies included have the redshift range 0.004 < z < 0.092, a large mass range (7.6 < log M * / M < 11.6), and star-formation rates of ∼ 10 −4 to ∼ 10 1 M yr −1 . For each galaxy, we include two spectral cubes and a set of spatially resolved 2D maps: single-and multicomponent emission-line fits (with dust extinction corrections for strong lines), local dust extinction and star-formation rate. Calibration of the fibre throughputs, fluxes and differential-atmosphericrefraction has been improved over the Early Data Release. The data have average spatial resolution of 2.16 arcsec (FWHM) over the 15 arcsec diameter field of view and spectral (kinematic) resolution R = 4263 (σ = 30 km s −1 ) around Hα. The relative flux calibration is better than 5% and absolute flux calibration better than ±0.22 mag, with the latter estimate limited by galaxy photometry. The data are presented online through the Australian Astronomical Observatory's Data Central.