In this paper we describe the survey design for the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) Cycle 1 JWST Treasury program, which executed its early imaging component in November 2022. The UNCOVER survey includes ultradeep (∼29-30AB) imaging of ∼45 arcmin^2 on and around the well-studied Abell 2744 galaxy cluster at z=0.308 and will follow-up ∼500 galaxies with extremely deep low-resolution spectroscopy with the NIRSpec/PRISM during the summer of 2023. We describe the science goals, survey design, target selection, and planned data releases. We also present and characterize the depths of the first NIRCam imaging mosaic, highlighting previously unparalleled resolved and ultradeep 2-4 micron imaging of known objects in the field. The UNCOVER primary NIRCam mosaic spans 28.8 arcmin^2 in seven filters (F115W, F150W, F200W, F277W, F356W, F410M, F444W) and 16.8 arcmin^2 in our NIRISS parallel (F115W, F150W, F200W, F356W, and F444W). To maximize early community use of the Treasury data set, we publicly release full reduced mosaics of public JWST imaging including 45 arcmin^2 NIRCam and 17 arcmin^2 NIRISS mosaics on and around the Abell 2744 cluster, including the Hubble Frontier Field primary and parallel footprints.