The <i>Gaia</i>-ESO Public Spectroscopic Survey: Motivation, implementation, GIRAFFE data processing, analysis, and final data products

Context. The Gaia-ESO Public Spectroscopic Survey is an ambitious project designed to obtain astrophysical parameters and elemental abundances for 100 000 stars, including large representative samples of the stellar populations in the Galaxy, and a well-defined sample of 60 (plus 20 archive) open clusters. We provide internally consistent results calibrated on benchmark stars and star clusters, extending across a very wide range of abundances and ages. This provides a legacy data set of intrinsic value, and equally a large wide-ranging dataset that is of value for the homogenisation of other and future stellar surveys and Gaia's astrophysical parameters. Aims. This article provides an overview of the survey methodology, the scientific aims, and the implementation, including a description of the data processing for the GIRAFFE spectra. A companion paper introduces the survey results. Methods. Gaia-ESO aspires to quantify both random and systematic contributions to measurement uncertainties. Thus, all available spectroscopic analysis techniques are utilised, each spectrum being analysed by up to several different analysis pipelines, with considerable effort being made to homogenise and calibrate the resulting parameters. We describe here the sequence of activities up to delivery of processed data products to the ESO Science Archive Facility for open use. Results. The Gaia-ESO Survey obtained 202 000 spectra of 115 000 stars using 340 allocated VLT nights between December 2011 and January 2018 from GIRAFFE and UVES. Conclusions. The full consistently reduced final data set of spectra was released through the ESO Science Archive Facility in late 2020, with the full astrophysical parameters sets following in 2022. A companion article reviews the survey implementation, scientific highlights, the open cluster survey, and data products.