A prosthetic heart valve is designed to be circumferentially collapsible for less invasive delivery into the patient. At the implant site the valve re-expands to a larger circumferential size, i.e., the size that it has for operation as a replacement for one of the patient's native heart valves. The valve includes structures that, at the implant site, extend radially outwardly to engage tissue structures above and below the native heart valve annulus. These radially outwardly extending structures clamp the native tissue between them and thereby help to anchor the prosthetic valve at the desired location in the patient.