We present the survey design, implementation, and outlook for COSMOS-Web, a 255 hr treasury program conducted by the James Webb Space Telescope in its first cycle of observations. COSMOS-Web is a contiguous 0.54 deg² NIRCam imaging survey in four filters (F115W, F150W, F277W, and F444W) that will reach 5σ point-source depths ranging ∼ 27.5-28.2 mag. In parallel, we will obtain 0.19 deg² of MIRI imaging in one filter (F770W) reaching 5σ point-source depths of ∼ 25.3-26.0 mag. COSMOS-Web will build on the rich heritage of multiwavelength observations and data products available in the COSMOS field. The design of COSMOS-Web is motivated by three primary science goals: (1) to discover thousands of galaxies in the Epoch of Reionization (6 ≲ z ≲ 11) and map reionization's spatial distribution, environments, and drivers on scales sufficiently large to mitigate cosmic variance, (2) to identify hundreds of rare quiescent galaxies at z > 4 and place constraints on the formation of the universe's most-massive galaxies (M(⋆) > 10^(10) M(⊙)), and (3) directly measure the evolution of the stellar-mass-to-halo-mass relation using weak gravitational lensing out to z ∼ 2.5 and measure its variance with galaxies' star formation histories and morphologies. In addition, we anticipate COSMOS-Web's legacy value to reach far beyond these scientific goals, touching many other areas of astrophysics, such as the identification of the first direct collapse black hole candidates, ultracool subdwarf stars in the Galactic halo, and possibly the identification of z > 10 pair-instability supernovae. In this paper we provide an overview of the survey's key measurements, specifications, goals, and prospects for new discovery.