The structure and phase behavior of liquid-mercury-supported molecular films of fatty acids (CH3(CH2)(n-2)COOH, denoted CnOOH) were studied for molecular lengths 7 less than or equal to n less than or equal to 24, by surface tensiometry and X-ray methods. Two qualitatively different film structures were found, depending on coverage. For high coverage, the film consists of a monolayer of roughly surface-normal molecules, showing a pressure-dependent sequence of structures similar, though not identical, to that of the corresponding water-supported Langmuir films. At low coverage, phases consisting of surface-parallel molecules are found, not observed on the aqueous subphases employed to date. In this range, a two-dimensional (2D) gas followed by a single and, for 14 less than or equal to n less than or equal to 24, also by a double layer of surface-parallel molecules is found as coverage is increased. Depending on chain length, the flat-lying phases have a crystalline 2D-ordered, a smectic-like 1D-ordered, or a disordered in-plane structure consisting of molecular dimers. The structure and thermodynamics of the films are discussed.