In order to understand the ammonia synthesis reaction over Ru-based catalysts, we have performed a series of density functional calculations of the adsorption and dissociation of N-2 on Ru(0001). We find molecularly and atomically adsorbed states of N-2 on Ru(0001) with structures, binding energies, and vibrational frequencies in good agreement with experimental data for these systems. We explain on the basis of the electronic structure of the Ru surface, the large difference in adsorption energy of N atoms in the hcp and fee sites, the large diffusion barrier, and the large indirect N-N repulsion. The reaction path is determined using the hyperplane adaptive constraint method. We find that the lowest energy path shows a sizable barrier toward dissociation of 130 kJ/mol. During dissociation, the molecule, which is standing perpendicular to the surface in the molecularly adsorbed state, is first rotated into a metastable, flat-lying molecular precursor and is then dissociated into two adjacent hcp adsorption sites. We discuss the electronic structure along the reaction path and show that the Nz induced dipole moment varies substantially. The variation of the dipole moment is used to discuss a possible model of the promoting effect of Cs on this reaction.