We present the Multi-Object Spectroscopy of Transient (MOST) Hosts survey. The survey is planned to run throughout the 5 yr of operation of the Dark Energy Spectroscopic Instrument (DESI) and will generate a spectroscopic catalog of the hosts of most transients observed to date, in particular all the supernovae observed by most public, untargeted, wide-field, optical surveys (Palomar Transient Factory, PTF/intermediate PTF, Sloan Digital Sky Survey II, Zwicky Transient Facility, DECAT, DESIRT). Science cases for the MOST Hosts survey include Type Ia supernova cosmology, fundamental plane and peculiar velocity measurements, and the understanding of the correlations between transients and their host-galaxy properties. Here we present the first release of the MOST Hosts survey: 21,931 hosts of 20,235 transients. These numbers represent 36% of the final MOST Hosts sample, consisting of 60,212 potential host galaxies of 38,603 transients (a transient can be assigned multiple potential hosts). Of all the transients in the MOST Hosts list, only 26.7% have existing classifications, and so the survey will provide redshifts (and luminosities) for nearly 30,000 transients. A preliminary Hubble diagram and a transient luminosity–duration diagram are shown as examples of future potential uses of the MOST Hosts survey. The survey will also provide a training sample of spectroscopically observed transients for classifiers relying only on photometry, as we enter an era when most newly observed transients will lack spectroscopic classification. The MOST Hosts DESI survey data will be released on a rolling cadence and updated to match the DESI releases. Dates of future releases and updates are available through the https://mosthosts.desi.lbl.gov website.
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