Adsorption of an H2 molecule on PdN clusters (N = 2-4, 7, 13, 19, and 55) is investigated using density functional theory with the hybrid PBE0 functional. Low-energy PdN isomers, taken from a large pool of candidate structures for all cluster sizes (except N = 55), are used in systematic searches for the most stable PdNH2 (molecular) and PdN2H (dissociative) adsorption complexes. Molecular adsorption of H2 is found to occur strictly at atop sites, with the strongest binding typically occurring at the site with the smallest coordination. Binding of dissociated H atoms occurs preferentially on 3-fold faces and on certain favorable edge sites, while binding at atop sites is unstable. Dissociative adsorption is energetically preferred to molecular adsorption for all cluster sizes. The dissociative adsorption energy decreases with cluster size, with pronounced variations due to cluster size effects for the smallest clusters. Adsorption reaction pathways are computed for cluster sizes up to N = 13. Molecular adsorption is found to be barrierless in all cases. Dissociative adsorption occurs without a barrier for the pathways studied for N = 7 and 13 and with a small barrier on the smaller clusters. Finally, lowest-energy pathways for the migration of a dissociated hydrogen atom between local minima on a cluster surface are computed for the Pd4, Pd7, and Pd13 clusters. Calculated migration barriers range from 0.05 to 0.25 eV.