There is a significant delay between the time when light hits the retina and the time of the consequent percept. It has been hypothesized that the visual system attempts to correct for this latency by generating a percept representative of the way the world probably is at the time the percept is elicited, rather than a percept of the recent past. Here we show that such a 'perceiving the present' hypothesis explains a number of classical geometrical illusions: the Hering, Orbison, Müller-Lyer, Double Judd, Poggendorff, Corner, and Upside-down-T illusions. Each stimulus is perceived as it would project in the next moment were the observer moving through the scene the stimulus probably represents. We also examine one general class of predictions made by the hypothesis, and report psychophysical experiments confirming the predictions.