The chemical mechanism by which nitrogenase enzymes catalyze the hydrogenation of N-2 (and other multiply bonded substrates) at the (NFe7MoS9)-Fe-c(homocitrate) active site (FeMo-co) is unknown, despite the accumulation of much data on enzyme reactivity and the influences of key amino acids surrounding FeMo-co. The mutual influences of H-2, substrates, and the inhibitor CO on reactivity are key experimental tests for postulated mechanisms. Fundamental to all aspects of mechanism is the accumulation of H atoms (from e(-) + H+) on FeMo-co, and the generation and influences of coordinated H2. Here, I argue that the first introduction of H is via a water chain terminating at water 679 (PDB structure 1M1N, Azotobacter vinelandii) to one of the mu(3)-S atoms (S3B) of FeMo-co. Next, using validated density functional calculations of a full chemical representation of FeMo-co and its connected residues (alpha-275(Cys), alpha-442(His)), I have characterized more than 80 possibilities for the coordination of up to three H atoms, and H-2, and H + H-2, on the S2A, Fe2, S2B, Fe6, S3B domain of FeMo-co, which is favored by recent targeted mutagenesis results. Included are calculated reaction profiles for movements of H atoms (between S and Fe, and between Fe and Fe), for the generation of Fe-H-2, for association and dissociation of Fe-H-2 at various reduction levels, and for H/H-2 exchange. This is new hydrogen chemistry on an unprecedented coordination frame, with some similarities to established hydrogen coordination chemistry, and with unexpected and unprecedented structures such as Fe(S)(3)(H-2)(2)(H) octahedral coordination. General principles for the hydrogen chemistry of FeMo-co include (1) the stereochemical mobility of H bound to mu(3)-S, (2) the differentiated endo- and exo- positions at Fe for coordination of H and/or H-2, and (3) coordinative allosteric influences in which structural and dynamic aspects of coordination at one Fe atom are affected by coordination at another Fe atom, and by H on S atoms. Evidence of end-differentiation in FeMo-co is described, providing a rationale for the occurrence of Mo. The reactivity results are discussed in the context of the Thorneley-Lowe scheme for nitrogenase reactions, and especially the scheme for the HID reaction (2H(+) + 2e(-) + D-2 -> 2HD), using a model containing an H-entry site and at least two coordinative sites on FeMo-co. I propose that S3B is the H-entry site, suggest details for the H+ shuttle to S3B and subsequent movement of H atoms around FeMo-co preparatory to the binding and hydrogenation of N2 and other substrates, and suggest how H could be transferred to an alkyne substrate. I propose that S2B (normally hydrogen bonded to alpha-195(His)) has a modulatory function and is not an H-entry site. Finally, the recent first experimental trapping of a hydrogenated intermediate with EPR and ENDOR characterization is discussed, leading to a consensual model for the intermediate.