The Legacy ExtraGalactic UV Survey (LEGUS) is a Cycle 21 Treasury program on the Hubble Space Telescope aimed at the investigation of star formation and its relation with galactic environment in nearby galaxies, from the scales of individual stars to those of ∼kiloparsec-size clustered structures. Five-band imaging from the nearultraviolet to the I band with the Wide-Field Camera 3 (WFC3), plus parallel optical imaging with the Advanced Camera for Surveys (ACS), is being collected for selected pointings of 50 galaxies within the local 12 Mpc. The filters used for the observations with the WFC3 are F275W (λ2704 A˚), F336W(λ3355 A˚), F438W(λ4325 A˚), F555W(λ5308 A˚), and F814W(ë8024 A˚); the parallel observations with the ACS use the filters F435W (λ4328 A˚), F606W(λ5921 A˚), and F814W(λ8057 A˚). The multiband images are yielding accurate recent (≲50 Myr) star formation histories from resolved massive stars and the extinction-corrected ages and masses of star clusters and associations. The extensive inventories of massive stars and clustered systems will be used to investigate the spatial and temporal evolution of star formation within galaxies. This will, in turn, inform theories of galaxy evolution and improve the understanding of the physical underpinning of the gas-star formation relation and the nature of star formation at high redshift. This paper describes the survey, its goals and observational strategy, and the initial scientific results. Because LEGUS will provide a reference survey and a foundation for future observations with the James Webb Space Telescope and with ALMA, a large number of data products are planned for delivery to the community.