Metasurfaces are planar structures that locally modify the polarization, phase, and amplitude of light in reflection or transmission, thus enabling lithographically patterned flat optical components with functionalities controlled by design. Transmissive metasurfaces are especially important, as most optical systems used in practice operate in transmission. Several types of transmissive metasurfaces have been realized, but with either low transmission efficiencies or limited control over polarization and phase. Here we show a metasurface platform based on high-contrast dielectric elliptical nano-posts which provides complete control of polarization and phase with sub-wavelength spatial resolution and experimentally measured efficiency ranging from 72% to 97%, depending on the exact design. Such complete control enables the realization of most free-space transmissive optical elements such as lenses, phase-plates, wave-plates, polarizers, beam-splitters, as well as polarization switchable phase holograms and arbitrary vector beam generators using the same metamaterial platform.