ESO is embarking on the construction of a complex and high performance dual feed system that will allow Phase Referenced Imaging and Microarcsecond Astrometry (PRIMA) on the VLT Interferometer on Cerro Paranal in Chile. In this paper, I will describe in some detail the scientific objectives of this facility that drive the technical specifications and justify the chosen priorities. Of particular importance because of its uniqueness and rich variety of scientific applications will be the early development of the components allowing very high precision astrometry on sources such as extrasolar planets, binaries in nearby clusters, microlenses in the halo, and stars in the circumnuclear cluster in the galactic center.