The GALEX Ultraviolet Virgo Cluster Survey (GUViCS) is a complete blind survey of the Virgo cluster covering ∼40 sq. deg in the far UV (FUV, λeff = 1539 Å Δλ = 442 Å) and ∼120 sq. deg in the near UV (NUV, λeff = 2316 Å, Δλ = 1060 Å). The goal of the survey is to study the ultraviolet (UV) properties of galaxies in a rich cluster environment, spanning a wide luminosity range from giants to dwarfs, and regardless of prior knowledge of their star formation activity. The UV data will be combined with those in other bands (optical: NGVS; far-infrared - submm: HeViCS; HI: ALFALFA) and with our multizone chemo-spectrophotometric models of galaxy evolution to make a complete and exhaustive study of the effects of the environment on the evolution of galaxies in high density regions. We present here the scientific objectives of the survey, describing the observing strategy and briefly discussing different data reduction techniques. Using UV data already in-hand for the central 12 sq. deg we determine the FUV and NUV luminosity functions of the Virgo cluster core for all cluster members and separately for early- and late-type galaxies and compare it to the one obtained in the field and other nearby clusters (Coma, A1367). This analysis shows that the FUV and NUV luminosity functions of the core of the Virgo clusters are flatter (α ∼ -1.1) than those determined in Coma and A1367. We discuss the possible origin of this difference.