Apparatus and methods are provided, including a mitral valve prosthesis for implantation at a native mitral valve complex of a subject. The prosthesis includes an inner support structure having a downstream section, and an upstream section, the upstream section having a cross-sectional area greater than the downstream section, the inner support structure being configured to be positioned at least partially on an atrial side of the native valve complex, and to apply an axial force directed toward a left ventricle. The prosthesis further includes an outer support structure having two or more engagement arms, the engagement arms being coupled to the inner support structure. The prosthesis is configured, upon implantation thereof, to clamp portions of leaflets of the native valve between the inner support structure and the engagement arms. Other embodiments are also described.
