Much of the central nervous system is involved in visuomotor transformations for goal-directed gaze and reach movements. These transformations are often described in terms of stimulus location, gaze fixation, and reach endpoints, as viewed through the lens of translational geometry. Here, we argue that the intrinsic (primarily rotational) 3-D geometry of the eye-head-reach systems determines the spatial relationship between extrinsic goals and effector commands, and therefore the required transformations. This approach provides a common theoretical framework for understanding both gaze and reach control. Combined with an assessment of the behavioral, neurophysiological, imaging, and neuropsychological literature, this framework leads us to conclude that (a) the internal representation and updating of visual goals are dominated by gaze-centered mechanisms, but (b) these representations must then be transformed as a function of eye and head orientation signals into effector-specific 3-D movement commands.