Density functional theory has been used to study the interaction of molecular and atomic hydrogen with (5,5) and (6,6) single-wall carbon nanotubes. Static calculations allowing for different degrees of structural relaxation are performed, in addition to dynamical simulations. Molecular physisorption inside and outside the nanotube walls is predicted to be the most stable state of those systems. The binding energies for physisorption of the H-2 molecule outside the nanotube are in the range 0.04-0.07 eV. This means that uptake and release of molecular hydrogen from nanotubes is a relatively easy process, as many experiments have proved. A chemisorption state, with the molecule dissociated and the two hydrogen atoms bonded to neighbor carbon atoms, has also been found. However, reaching this dissociative chemisorption state for an incoming molecule, or starting from the physisorbed molecule, is difficult because of the existence of a substantial activation barrier. The dissociative chemisorption deforms the tube and weakens the C-C bond. This effect can catalyze the shattering and scission of the tube by incoming hydrogen molecules with sufficient kinetic energy.