This paper presents an investigation of the behavior of rich hydrogen-air flames near the flammability limit. Ignition processes arc simulated using a computational model which involves the solution of the governing equations (for one-dimensional geometries) by implicit methods on an adaptive non-uniform grid, using detailed chemistry and a multi-species transport model. The reaction mechanism consists of 37 elementary reactions and 9 species. Calculations have been performed for different one-dimensional geometries. An investigation of the fundamental flammability limit {intrinsic to the combustion system itself), which is governed only by the physical and chemical processes in the gaseous mixture, is carried out by eliminating external factors such as heat loss, buoyancy, etc, in the calculations. Compulations have been performed for hydrogen-air mixtures of varying hydrogen content. Mixtures containing 75% or less hydrogen are found to be steadily propagating, whereas mixtures containing 82% or more hydrogen extinguish. Inbetween these bounds, an oscillatory flame propagation is observed, which is characterized by an oscillation between a fast and a slow flame propagation. Furthermore, the effect of the pressure on the flammability limit is studied