Pigeons categorized a moving target in terms of its speed and direction in an adaptation of the randomization procedure used to study human categorization behavior (Ashby & Maddox, 1998). The target moved according to vectors that were sampled with equal probabilities from two slightly overlapping bivariate normal distributions with the dimensions of speed and direction. On the average, pigeons categorized optimally in that they attended to either speed or direction alone, or divided attention between them, as was required by different reinforcement contingencies. Decision bounds were estimated for individual pigeons for each attentional task. Average slopes and y intercepts of these individually estimated decision bounds closely approximated the corresponding values for optimal decision bounds. There is therefore at least one task in which pigeons, on the average, display flexibility and quantitative precision in allocating attention to speed and direction when they categorize moving targets.