We report a CO(J = 1-0) aperture-synthesis survey of the central regions of 20 nearby spiral galaxies. The sample is selected on the basis of inclination, single-dish CO flux, and lack of significant dynamical perturbation. No selection is made on the basis of their nuclear activity, starburst, or infrared luminosity. The observations have been made using the Nobeyama and Owens Valley millimeter arrays, with typical resolutions of 4'' (300 pc) and 20 km s^-1, and sensitivities of ~30 mJy beam^-1 for a 20 km s^-1 channel. In this paper, we present integrated intensity and mean velocity maps, azimuthally averaged radial distributions, and position-velocity plots. Most galaxies in the sample have scale lengths for the radial CO distribution that are much smaller in the nuclear regions than in the outer disks. Typically, the CO emission falls to 1/e of the central peak value at radii of ~500 pc. This is significantly smaller than the scale lengths for the global CO distributions in the galactic disks. CO position-velocity diagrams exhibit a steep rise in rotation velocity, dV/dr ~ 1 km s^-1 pc^-1, in the central regions of most galaxies. The CO line width typically reaches ~95% of the overall H I line width within a radius of 1 kpc. This steep rise in the rotation velocity mainly reflects the highly concentrated mass distributions in the galactic centers, though it may well be partly due to noncircular motions in barred galaxies. In a number of galaxies, the position-velocity diagrams show a small central hole (d 100 pc) in CO emission. In many galaxies, we detect CO clumps of subkiloparsec size which may be giant molecular associations (GMAs). In the Appendix, we demonstrate that small holes at the galactic center can be readily detected in position-velocity diagrams even when they are smaller than the spatial resolution of the data.