Sex differences in cerebral organization and functioning, and the apparent superiority of males in spatial ability are examined in this paper. Attention is given to several kinds of cognitive and perceptual tasks in which sex differences in spatial ability have been shown to exist; among these are tasks involving: (1) recall and detection of shapes, (2) geometry and mathematics, (3) directional sense, (4) Piagetian skills and (5) the game of chess. A neurological model for sex differences in spatial ability is discussed, which suggests that some brains are further specialized (lateralized) for spatial analysis than others, and that these "further specialized" brains are more frequently male than female. Medical research with war veterans who have suffered brain injuries, data from anatomical and clinical studies and results of testing of normal individuals are cited to suggest that the left hemisphere of the brain seems primarily organized for verbal function, the right hemisphere for visual-spatial functions. An alternative model for sex differences in spatial ability suggests that women prefer to code information phonologically (due to their earlier language development) and that men prefer to code information visually. Data are presented which support this view. The possibilities of evolutionary selection for male spatial superiority and the effects of sex steroid hormones on brain specialization and nervous system activity are considered. (BRT)