We present PHANGS-ALMA, the first survey to map CO J=2-1 line emission at ~1" ~100pc spatial resolution from a representative sample of 90 nearby (d<~20Mpc) galaxies that lie on or near the z=0 "main sequence" of star-forming galaxies. CO line emission traces the bulk distribution of molecular gas, which is the cold, star-forming phase of the interstellar medium. At the resolution achieved by PHANGS-ALMA, each beam reaches the size of a typical individual giant molecular cloud, so that these data can be used to measure the demographics, life cycle, and physical state of molecular clouds across the population of galaxies where the majority of stars form at z=0. This paper describes the scientific motivation and background for the survey, sample selection, global properties of the targets, Atacama Large Millimeter/submillimeter Array (ALMA) observations, and characteristics of the delivered data and derived data products. As the ALMA sample serves as the parent sample for parallel surveys with MUSE on the Very Large Telescope, the Hubble Space Telescope, AstroSat, the Very Large Array, and other facilities, we include a detailed discussion of the sample selection. We detail the estimation of galaxy mass, size, star formation rate, CO luminosity, and other properties, compare estimates using different systems and provide best-estimate integrated measurements for each target. We also report the design and execution of the ALMA observations, which combine a Cycle 5 Large Program, a series of smaller programs, and archival observations. Finally, we present the first 1" resolution atlas of CO emission from nearby galaxies and describe the properties and contents of the first PHANGS-ALMA public data release.