An implantable closure structure is delivered using minimally invasive techniques, and inhibits the migration of liquid and particulate matter from inside a physiological cavity or opening, such as an aneurysm or a septal defect, as well as inhibiting the flow of liquid and particulate matter, such as from an associated blood vessel or chamber, into the physiological cavity or opening. The device has a closure structure that covers the neck or opening of a cavity and has one or more anchoring structures for supporting and retaining the closure structure in place across the cavity or opening.
