A circulation control valve includes an opening between the valve exterior and an interior passage, an internal closure device for permitting and preventing flow through the opening, a valve device initially preventing flow through the opening, and an internal chamber. The valve device opens upon application of a pressure differential between the passage and the exterior to thereby permit communication through the opening, and the closure device displaces upon a second pressure differential between the passage and the internal chamber to thereby prevent communication through the opening. Another valve includes first and second valve devices. Communication through the opening is permitted upon application of the first pressure differential to the first device, thereby unbalancing a first piston, and fluid communication through the opening is prevented upon application of the second pressure differential to the second device, thereby unbalancing a second piston having a greater piston area than the first piston.