Apparatus for use with a native valve of a heart of a subject, the apparatus comprising: an implant (140, 240, 260, 380, 460) comprising: a support frame, having a compressed state, and an expanded state in which the support frame defines an opening (150', 390) therethrough, and is dimensioned to be placed against an upstream surface of the native valve such that the opening (150', 390) is disposed over an orifice defined by the native valve; a flexible sheet (146, 246, 266a, 386, 466); and a valve frame (144, 244, 264, 384, 464): having a compressed state, and an expanded state in which the valve frame (144, 244, 264, 384, 464) defines a lumen (30, 250) therethrough, and comprising a valve member disposed within the lumen (30, 250); and a catheter for percutaneously delivering the implant (140, 240, 260, 380, 460) in a compressed state thereof to the native valve, with the support frame in the compressed state and the valve frame (144, 244, 264, 384, 464) in the compressed state; characterized in that: the valve frame (144, 244, 264, 384, 464) is coupled to the support frame via the flexible sheet (146, 246, 266a, 386, 466) such that when the support frame is in its expanded state, and the valve frame (144, 244, 264, 384, 464) is in its expanded state, at least part of the lumen (30, 250) is disposed within the opening (150', 390), and the valve frame (144, 244, 264, 384, 464) is not in contact with the support frame; and the implant (140, 240, 260, 380, 460) is disposed relative to the catheter for implanting transfemorally, with the valve frame (144, 244, 264, 384, 464) closer to a distal end of the catheter than the support frame is to the distal end of the catheter. an implant (140, 240, 260, 380, 460) comprising: a support frame, having a compressed state, and an expanded state in which the support frame defines an opening (150', 390) therethrough, and is dimensioned to be placed against an upstream surface of the native valve such that the opening (150', 390) is disposed over an orifice defined by the native valve; a flexible sheet (146, 246, 266a, 386, 466); and a valve frame (144, 244, 264, 384, 464): having a compressed state, and an expanded state in which the valve frame (144, 244, 264, 384, 464) defines a lumen (30, 250) therethrough, and comprising a valve member disposed within the lumen (30, 250); and a support frame, having a compressed state, and an expanded state in which the support frame defines an opening (150', 390) therethrough, and is dimensioned to be placed against an upstream surface of the native valve such that the opening (150', 390) is disposed over an orifice defined by the native valve; a flexible sheet (146, 246, 266a, 386, 466); and a valve frame (144, 244, 264, 384, 464): having a compressed state, and an expanded state in which the valve frame (144, 244, 264, 384, 464) defines a lumen (30, 250) therethrough, and comprising a valve member disposed within the lumen (30, 250); and having a compressed state, and an expanded state in which the valve frame (144, 244, 264, 384, 464) defines a lumen (30, 250) therethrough, and comprising a valve member disposed within the lumen (30, 250); and a catheter for percutaneously delivering the implant (140, 240, 260, 380, 460) in a compressed state thereof to the native valve, with the support frame in the compressed state and the valve frame (144, 244, 264, 384, 464) in the compressed state; characterized in that: the valve frame (144, 244, 264, 384, 464) is coupled to the support frame via the flexible sheet (146, 246, 266a, 386, 466) such that when the support frame is in its expanded state, and the valve frame (144, 244, 264, 384, 464) is in its expanded state, at least part of the lumen (30, 250) is disposed within the opening (150', 390), and the valve frame (144, 244, 264, 384, 464) is not in contact with the support frame; and the implant (140, 240, 260, 380, 460) is disposed relative to the catheter for implanting transfemorally, with the valve frame (144, 244, 264, 384, 464) closer to a distal end of the catheter than the support frame is to the distal end of the catheter. the valve frame (144, 244, 264, 384, 464) is coupled to the support frame via the flexible sheet (146, 246, 266a, 386, 466) such that when the support frame is in its expanded state, and the valve frame (144, 244, 264, 384, 464) is in its expanded state, at least part of the lumen (30, 250) is disposed within the opening (150', 390), and the valve frame (144, 244, 264, 384, 464) is not in contact with the support frame; and the implant (140, 240, 260, 380, 460) is disposed relative to the catheter for implanting transfemorally, with the valve frame (144, 244, 264, 384, 464) closer to a distal end of the catheter than the support frame is to the distal end of the catheter.