We present the results of a new VLA H I Imaging survey of Virgo galaxies, the VLA Imaging survey of Virgo galaxies in Atomic gas (VIVA). The survey includes high-resolution H I data of 53 carefully selected late type galaxies (48 spirals and five irregular systems). The goal is to study environmental effects on H I gas properties of cluster galaxies to understand which physical mechanisms affect galaxy evolution in different density regions, and to establish how far out the impact of the cluster reaches. As a dynamically young cluster, Virgo contains examples of galaxies experiencing a variety of environmental effects. Its nearness allows us to study each galaxy in great detail. We have selected Virgo galaxies with a range of star formation properties in low to high density regions (at projected distances from M87, d ₈₇ = 0.3-3.3 Mpc). Contrary to previous studies, more than half of the galaxies in the sample (~60%) are fainter than 12 mag in B_T . Overall, the selected galaxies represent the late type Virgo galaxies (S0/a to Sd/Irr) down to m_p 14.6 fairly well in morphological type, systemic velocity, subcluster membership, H I mass, and deficiency. The H I observations were done in C short (CS) configuration of the VLA radio telescope, with a typical spatial resolution of 15'' and a column density sensitivity of 3-5 × 10¹⁹ cm^–2 in 3σ per 10 km s^–1 channel. The survey was supplemented with data of comparable quality from the NRAO archive, taken in CS or C configuration. In this paper, we present H I channel maps, total intensity maps, velocity fields, velocity dispersions, global/radial profiles, position-velocity diagrams and overlays of H I/1.4 GHz continuum maps on the optical images. We also present H I properties such as total flux (S _{H I} ), H I mass (M _{H I} ), linewidths (W ₂₀ and W ₅₀), velocity (V _{H I} ), deficiency (def _{H I} ), and size (D ^eff _{H I} and D ^iso _{H I} ), and describe the H I morphology and kinematics of individual galaxies in detail. The survey has revealed details of H I features that were never seen before. In this paper, we briefly discuss differences in typical H I morphology for galaxies in regions of different galaxy densities. We confirm that galaxies near the cluster core (d ₈₇ 0.5 Mpc) have H I disks that are smaller compared to their stellar disks (D _{H I} /D ₂₅ < 0.5). Most of these galaxies in the core also show gas displaced from the disk, which is either currently being stripped or falling back after a stripping event. At intermediate distances (d ₈₇ ~ 1 Mpc) from the center, we find a remarkable number of galaxies with long one-sided H I tails pointing away from M87. In a previous letter, we argue that these galaxies are recent arrivals, falling into the Virgo core for the first time. In the outskirts, we find many gas-rich galaxies, with gas disks extending far beyond their optical disks. Interestingly, we also find some galaxies with H I disks that are smaller compared to their stellar disks at large clustercentric distances.