The development of a detailed, time-dependent, three-dimensional flame model (FLAME3D) for premixed laminar flames is reported in this paper. The model has been developed directly for and runs efficiently on several MIMD parallel computers. FLAME3D was used to study zero gravity, upward-and downward-propagating flames in a 9.5% hydrogen-air mixture. Multidimensional cellular flame structures are observed in zero gravity. This cellular flame grows and splits into multiple cells. The growth of the cellular structure is faster in the case of an upward-propagating flame. For the downward-propagating flame, the thermo-diffusive instability mechanism and the Rayleigh-Taylor mechanism oppose each other, resulting in a planar flame. The zero gravity cellular structures in a two-dimensional calculation grow slowly and do not undergo cell splitting, unlike in the three-dimensional simulation. This result indicates the need for studying complex flame phenomena using three-dimensional models such as the one discussed here.