JADES Data Release 4 Paper I: Sample Selection, Observing Strategy and Redshifts of the complete spectroscopic sample

This paper accompanies Data Release 4 of the JWST Deep Extragalactic Survey (JADES), which presents the full NIRSpec spectroscopy of the survey. We provide spectra of 5190 targets across GOODS-North and GOODS-South (including the Hubble Ultra Deep Field), observed with the low-dispersion (R $\sim$ 30-300) prism and three medium-resolution (R $\sim$ 1000) gratings spanning 0.8 $<\lambda<$ 5.5 microns; 2654 were also observed with the higher-resolution (R $\sim$ 2700) G395H grating. The tiered survey design obtained more than 20 hr exposures for $\sim$ 700 galaxies in the Deep and Ultra Deep tiers, and shallower observations ($\sim$ 1-3 hr per setting) of $>$ 4400 galaxies in the Medium tiers. Targets were selected from photometric redshifts or colours, with priority given to rest-UV-selected galaxies at $z>5.7$ and F444W-selected galaxies at $1.5<z<5.7$. We describe the full target selection and present spectroscopic redshifts and success rates. In total we obtain robust redshifts for 3297 galaxies, including 396 at $z>5.7$ and 2545 at $1.5<z<5.7$. To facilitate uniform analyses, we define'gold'sub-samples based on UV- and F444W-selection. Using the parent samples and redshift success rates, we construct rest-UV luminosity functions at $6 \lesssim z \lesssim 9$ from the Medium- and Deep-JWST tiers. Our number densities agree well with previous determinations from both photometric and spectroscopic samples, with modest interloper fractions confirming the reliability of photometric UV-bright galaxy selections at these redshifts.