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The experimental absorption line widths, for nuclei with spin 1/2, at nuclear magnetic resonance are given as a function of temperature for a number of molecular crystals. Temperatures ranged from 90°K to the melting points of the compounds. In some cases it has been possible to relate observed line structure and transitions in the line width to the existence and frequency of certain types of hindered rotational motion in the solid state. These deductions are based on mathematical considerations of the quantitative effect of such motions on the structure and second moment of an absorption line. It is emphasized that relatively low frequency motion of the order 105 cycles/second suffices to narrow the width of an absorption line from its value in the absence of that motion. 1,2-dichloroethane, 1,1,1-trichloroethane, and perfluoroethane were found to have line-width transitions coinciding with changes in crystal form and anomalies in the heat capacity. In 1,2-dichloroethane and perfluoroethane these transitions correspond to rotational motion about the long axis of the molecule. 1,1,1-trichloroethane has a line-width transition corresponding to this type of motion, but the heat capacity anomaly and change in crystal form coincides with a further small decrease in the line width to that characteristic of the liquid. A number of molecules including acetonitrile, methyl iodide, nitromethane, dimethyl mercury, and ammonia have absorption lines at 90°K corresponding to molecular rotation about the C3 figure axis. Various other data and interpretations are presented for 2,2-dimethyl propane, methanol, ethanol, acetone, methylamine, and the ethyl halides. The possibility of estimating the potential barriers hindering rotation in solids from the line-width transitions is discussed.