We present "The HI Nearby Galaxy Survey (THINGS)," a high spectral (<=5.2km/s) and spatial (~6") resolution survey of HI emission in 34 nearby galaxies obtained using the NRAO Very Large Array (VLA). The overarching scientific goal of THINGS is to investigate fundamental characteristics of the interstellar medium (ISM) related to galaxy morphology, star formation, and mass distribution across the Hubble sequence. Unique characteristics of the THINGS database are the homogeneous sensitivity as well as spatial and velocity resolution of the HI data, which is at the limit of what can be achieved with the VLA for a significant number of galaxies. A sample of 34 objects at distances 2<~D>~15Mpc (resulting in linear resolutions of ~100 to 500pc) are targeted in THINGS, covering a wide range of star formation rates (~10^-3^ to 6M_{sun}_/yr), total HI masses MHI (0.01 to 14x10^9^M_{sun}_), absolute luminosities MB (-11.5 to -21.7mag), and metallicities (7.5 to 9.2 in units of 12+log[O/H]). We describe the setup of the VLA observations, the data reduction procedures, and the creation of the final THINGS data products. We present an atlas of the integrated HI maps, the velocity fields, the second moment (velocity dispersion) maps and individual channel maps of each THINGS galaxy. The THINGS data products are made publicly available through a dedicated webpage (http://www.mpia.de/THINGS). Accompanying THINGS papers (in this issue of the Astronomical Journal) address issues such as the small-scale structure of the ISM, the (dark) matter distribution in THINGS galaxies, and the processes leading to star formation.