We have mapped the structure and connectivity of the Shapley supercluster core and its surroundings as traced by its member galaxies. To achieve this, we have carried out a new redshift survey of galaxies across a 21 square degree region (17×17 Mpc 2) of the Shapley supercluster containing 11 rich clusters, all lying at z ∼ 0.048. This survey aims to provide full spectroscopic coverage of galaxies for the Shapley Supercluster Survey (ShaSS), a multiwavelength survey based on high-quality optical ugri and near-infrared K-band imaging acquired with the 2.6-m VST and 4.1-m VISTA telescopes. Targets were selected from the VST images as having i < 18.0 and WISE/W1 < 15.5 mag, and the observations were carried out with the AAOmega spectrograph on the 3.9-m Anglo-Australian Telescope. We present here redshift measurements for 4027 galaxies, of which 957 belong to the Shapley Supercluster. Including existing measurements, there are now reliable redshifts for 95 per cent of galaxies within the survey's magnitude limits over the 21 deg 2 region. This has enabled us to: (i) obtain a detailed map of the galaxy number density across the whole region; (ii) identify new groups and cluster substructures; (iii) establish the existence of a stream of galaxies connecting A 3559 to the supercluster core. We provide updated central redshifts and velocity dispersions of the 11 clusters, confirming that they all lie within 1300 km s −1 of the central cluster Abell 3558. These 11 systems are all interconnected and lie within a coherent sheet of galaxies that fills the entire survey region without gaps. Clear velocity caustics extend right to the survey boundary, indicating that the entire structure is gravitationally bound and in the process of collapse.