A catheter system and method for achieving total cardiopulmonary bypass during heart surgery. A venous perfusion catheter is inserted peripherally into a preselected vein where it is advanced and positioned at the atrio-caval junction. The venous perfusion catheter has first and second balloons which when inflated respectively occlude the inferior and superior vena cava thereby precluding blood flow into the right atrium. An arterial perfusion catheter is inserted peripherally into a preselected arterial vessel and advanced within the vessel and positioned in the ascending aorta cephalid of the junction of the coronary arteries with the aortic root. A second flexible arterial cannula is mounted in sliding relationship with the first flexible cannula and carries an inflatable balloon adjacent its distal end to provide for occlusion of the ascending aorta. A first flexible cannula has a first lumen and an arterial venting orifice communicating with the first lumen defining a single flow path for the passage of cardioplegia solution to arrest the heart or for the evacuation of blood from the aortic root. A third lumen extends axially through the first flexible arterial cannula and communicates with a multiplicity of openings in the distal tip of the cannula defining a flow path for suctioning blood from the left ventricle. The second flexible cannula of the arterial perfusion catheter has a first cavity extending axially therethrough that communicates with an opening at its distal tip to permit the passage of blood delivered by the cardiopulmonary bypass pump into arterial circulation.