A system for delivering a collapsible and re-expandable prosthetic heart valveinto a patient includes a valve support structure (140) around which the valve(10) is disposed in a collapsed condition. A sheath structure (110) surroundsthe collapsed valve, but can be moved relative to the valve to uncover it for expansionat the desired implant site in the patient. The sheath structure may be variouslymounted and moved to deploy the valve in various ways. For example, the sheathstructure may include multiple parts, each of which can be moved separately toseparately deploy various parts of the valve. The apparatus may have other aspects,such as the ability to reverse deployment of the valve, the ability to pass otherinstrumentation through the valve delivery apparatus, the ability to be smoothlywithdrawn from the patient after deployment of the valve, etc.