The mechanism of positional localization has recently been debated due to interest in the flash-lag effect, which occurs when a briefly flashed stationary stimulus is perceived to lag behind a spatially aligned moving stimulus. Here we report positional localization observed at motion offsets as well as at onsets. In the ‘flash-lead’ effect, a moving object is perceived to be behind a spatially concurrent stationary flash before the two disappear. With ‘reverse-repmo’, subjects mis-localize the final position of a moving bar in the direction opposite to the trajectory of motion. Finally, we demonstrate that simultaneous onset and offset effects lead to a perceived compression of visual space. By characterizing illusory effects observed at motion offsets as well as at onsets, we provide evidence that the perceived position of a moving object is the result of an averaging process over a short time period, weighted towards the most recent positions. Our account explains a variety of motion illusions, including the compression of moving shapes when viewed through apertures.