We present new VLA C+D-array HI observations and optical and NIR imaging of the well known interacting system NGC 4038/9, ``The Antennae. The radio data reveal a wealth of gaseous sub-structure both within the main bodies of the galaxies and along the tidal tails. In agreement with previous HI studies, we find that the northern tail has HI along its outer length, but none along its base. We suggest that the HI at the base of this tail has been ionized by massive stars in the disk of NGC 4038. The gas in the southern tail has a bifurcated structure, with one filament lying along the optical tail and another running parallel to it but with no optical counterpart. The two filaments join just before the location of several star forming regions near the end of the tail. The HI velocity field at the end of the tail is dominated by strong velocity gradients which suggest that at this location the tail is bending away from us. We delineate and examine two regions within the tail previously identified as possible sites of a so-called ``tidal dwarf galaxy condensing out of the expanding tidal material. The tail velocity gradients mask any clear kinematic signature of a self-gravitating condensation in this region. A dynamical analysis suggest that there is not enough mass in gas alone for either of these regions to be self-gravitating. Conversely, if they are bound they require a significant contribution to their dynamical mass from evolved stars or dark matter. (Abridged)