The Large and Small Magellanic Clouds are unique local laboratories for studying the formation and evolution of small galaxies in exquisite detail. The Survey of the MAgellanic Stellar History (SMASH) is an NOAO community Dark Energy Camera (DECam) survey of the Clouds mapping 480 deg2 (distributed over ∼2400 square degrees at ∼20% filling factor) to ∼24th mag in ugriz. The primary goals of SMASH are to identify low surface brightness stellar populations associated with the stellar halos and tidal debris of the Clouds, and to derive spatially resolved star formation histories. Here, we present a summary of the survey, its data reduction, and a description of the first public Data Release (DR1). The SMASH DECam data have been reduced with a combination of the NOAO Community Pipeline, the PHOTRED automated point-spread-function photometry pipeline, and custom calibration software. The astrometric precision is ∼15 mas and the accuracy is ∼2 mas with respect to the Gaia reference frame. The photometric precision is ∼0.5%–0.7% in griz and ∼1% in u with a calibration accuracy of ∼1.3% in all bands. The median 5σ point source depths in ugriz are 23.9, 24.8, 24.5, 24.2, and 23.5 mag. The SMASH data have already been used to discover the Hydra II Milky Way satellite, the SMASH 1 old globular cluster likely associated with the LMC, and extended stellar populations around the LMC out to R ∼ 18.4 kpc. SMASH DR1 contains measurements of ∼100 million objects distributed in 61 fields. A prototype version of the NOAO Data Lab provides data access and exploration tools.