The positional specificity of short-term visual memory for a variety of 3-D shapes was investigated in a series of ‘same’/‘different’ discrimination experiments, with computer-rendered stimuli displayed either at the same or at different locations in the visual field. For animal-like shapes, we found complete translation invariance, regardless of the interstimulus similarity, and irrespective of direction and size of the displacement (experiments 1 and 2). Invariance to translation was obtained also with animal-like stimuli that had been ‘scrambled’ by randomizing the relative locations of their parts (experiment 3). The invariance broke down when the stimuli were made to differ in their composition, but not in the shapes of the corresponding parts (experiments 4 and 5). We interpret this pattern of findings in the context of several current theories of recognition, focusing in particular on the issue of the representation of object structure.