We describe the VISTA Science Archive (VSA) and its first public release of data from five of the six VISTA public surveys. The VSA exists to support the VISTA surveys through their lifecycle: the VISTA Public Survey consortia can use it during their quality control assessment of survey data products before submission to the ESO Science Archive Facility (ESO SAF); it supports their exploitation of survey data prior to its publication through the ESO SAF; and, subsequently, it provides the wider community with survey science exploitation tools that complement the data product repository functionality of the ESO SAF. This paper has been written in conjunction with the first public release of public survey data through the VSA and is designed to help its users understand the data products available and how the functionality of the VSA supports their varied science goals. We describe the design of the database and outline the database-driven curation processes that take data from nightly pipeline-processed and calibrated FITS files to create science-ready survey datasets. Much of this design, and the codebase implementing it, derives from our earlier WFCAM Science Archive (WSA), so this paper concentrates on the VISTA-specific aspects and on improvements made to the system in the light of experience gained in operating the WSA.