We consider the adsorption properties of superfluid \(^4\)He films on carbon nanotubes. One major factor in the adsorption is the surface tension force arising from the very small diameter of the nanotubes. Calculations show that surface tension keeps the film thickness on the tubes very thin even when the helium vapor is increased to the saturated pressure. The weakened Van der Waals force due to the cylindrical geometry also contributes to this. Both of these effects act to lower the predicted velocity of third sound propagation along the tubes. It does not appear that superfluidity will be possible on single-walled nanotubes of diameter about one nm, since the film thickness is less than 3 atomic layers even at saturation. Superfluidity is possible on larger-diameter nanotube bundles and multi-walled nanotubes, however. We have observed third sound signals on nanotube bundles of average diameter 5 nm which are sprayed onto a Plexiglass surface, forming a network of tubes.