This paper presents a complete description of Virgo, the French-Italian gravitational wave detector. The detector, built at Cascina, near Pisa (Italy), is a very large Michelson interferometer, with 3 km-long arms. In this paper, following a presentation of the physics requirements, leading to the specifications for the construction of the detector, a detailed description of all its different elements is given. These include civil engineering infrastructures, a huge ultra-high vacuum (UHV) chamber (about 6000 cubic metres), all of the optical components, including high quality mirrors and their seismic isolating suspensions, all of the electronics required to control the interferometer and for signal detection. The expected performances of these different elements are given, leading to an overall sensitivity curve as a function of the incoming gravitational wave frequency. This description represents the detector as built and used in the first data-taking runs. Improvements in different parts have been and continue to be performed, leading to better sensitivities. These will be detailed in a forthcoming paper. © 2012 IOP Publishing Ltd and Sissa Medialab srl.