In this work, adsorption of hydrogen on Al-13 clusters has been investigated theoretically using the density functional theory (DFT) approach. We have performed geometry optimization of atomic and molecular hydrogen in the proximity of Al-13 and calculated the binding energy and electronic properties of the stable Al-13 + H-n assemblies. We have also calculated the energy barrier for the hydrogen atom transition between different adsorption sites on Al-13 cluster as well as the activation energy for the dissociation and adsorption of molecular hydrogen. We found that the hydrogen atom adsorbs on the surface of Al-13 cluster, without an energy barrier onto atop, bridge and hollow sites. A small barrier for H atom transition from one adsorption site to another together with the minor energy difference between the most stable isomers points towards high mobility of the hydrogen atom on the surface. The calculated dissociation-adsorption barrier for the hydrogen molecule of similar to 14 kcal/mol and a desorption barrier of similar to 19 kcal/mol together with a high theoretical storage capacity of Al-13 clusters suggest further investigations of Al nanostructures for application in hydrogen storage.