Apparatus and methods are provided including a mitral valve prosthesis. The prosthesis includes an inner support structure having downstream and upstream sections, the upstream section having a cross-sectional area greater than the downstream section. The inner support structure is configured to be positioned on an atrial side of the native valve complex, and to prevent the prosthesis from being dislodged into the left ventricle. A prosthetic valve having prosthetic valve leaflets is coupled to the inner support structure. An outer support structure has engagement arms, downstream ends of the engagement arms being coupled to the inner support structure. The prosthesis is configured such that, upon implantation thereof: downstream ends of the native valve leaflets, downstream ends of the engagement arms, and downstream ends of the prosthetic leaflets, are disposed at a longitudinal distance from one another of less than 3 mm. Other embodiments are also described.