The interaction of atomic and molecular hydrogen with the Ag(111) surface is studied using periodic density functional total-energy calculations. This paper focuses on the site preference for adsorption, ordered structures, and energy barriers for H diffusion and H recombination. Chemisorbed H atoms are unstable with respect to the H(2) molecule in all adsorption sites below monolayer coverage. The three-hollow sites are energetically the most favorable for H chemisorption. The binding energy of H to the surface decreases slightly up to one monolayer, suggesting a small repulsive H-H interaction on nonadjacent sites. Subsurface and vacancy sites are energetically less favorable for H adsorption than on-top sites. Recombination of chemisorbed H atoms leads to the formation of gas-phase H(2) with no molecular chemisorbed state. Recombination is an exothermic process and occurs on the bridge site with a pronounced energy barrier. This energy barrier is significantly higher than that inferred from experimental temperature-programmed desorption (TPD) studies. However, there is significant permeability of H atoms through the recombination energy barrier at low temperatures, thus increasing the rate constant for H(2) desorption due to quantum tunneling effects, and improving the agreement between experiment and theory.