  The Dark Energy Spectroscopic Instrument (DESI) was designed to conduct a
survey covering 14,000 deg$^2$ over five years to constrain the cosmic
expansion history through precise measurements of Baryon Acoustic Oscillations
(BAO). The scientific program for DESI was evaluated during a five month Survey
Validation (SV) campaign before beginning full operations. This program
produced deep spectra of tens of thousands of objects from each of the stellar
(MWS), bright galaxy (BGS), luminous red galaxy (LRG), emission line galaxy
(ELG), and quasar target classes. These SV spectra were used to optimize
redshift distributions, characterize exposure times, determine calibration
procedures, and assess observational overheads for the five-year program. In
this paper, we present the final target selection algorithms, redshift
distributions, and projected cosmology constraints resulting from those
studies. We also present a `One-Percent survey' conducted at the conclusion of
Survey Validation covering 140 deg$^2$ using the final target selection
algorithms with exposures of a depth typical of the main survey. The Survey
Validation indicates that DESI will be able to complete the full 14,000 deg$^2$
program with spectroscopically-confirmed targets from the MWS, BGS, LRG, ELG,
and quasar programs with total sample sizes of 7.2, 13.8, 7.46, 15.7, and 2.87
million, respectively. These samples will allow exploration of the Milky Way
halo, clustering on all scales, and BAO measurements with a statistical
precision of 0.28% over the redshift interval $z<1.1$, 0.39% over the redshift
interval $1.1<z<1.9$, and 0.46% over the redshift interval $1.9<z<3.5$.
