We use the term clairvoyant to refer to networks that provide on-demand visibility for any flow at any time. Traditionally, network visibility is achieved by instrumenting and passively monitoring all flows in a network. SDN networks, by design endowed with full visibility, offer another alternative to network-wide flow monitoring. Both approaches incur significant capital and operational costs to make networks clairvoyant. In this paper, we argue that we can make any existing network clairvoyant by installing one or more SDN-enabled switches and a specialized controller to support on-demand visibility. We analyze the benefits and costs of such clairvoyant networks and provide a basic design by integrating two existing mechanisms for updating paths through legacy switches with SDN, telekinesis and magnet MACs. Our evaluation on a lab testbed and through extensive simulations show that, even with a single SDN-enabled switch, operators can make any flow visible for monitoring within milliseconds, albeit at 38% average increase in path length. With as many as 2% strategically chosen legacy switches replaced with SDN switches, clairvoyant networks achieve on-demand flow visibility with negligible overhead.