A prosthetic device (300; 400) for treating heart valve regurgitation at a native mitral valve of a heart, the prosthetic device (300; 400) comprising: a flexible sheet comprising blood-impermeable material and having an upper end portion (302; 410) and a free end portion (306; 412) extending away from the upper end portion (302; 406); means for securing the upper end portion (302; 406) of the sheet to an annulus of the native mitral valve above a first leaflet of the native mitral valve, such that the device (300; 400) is implanted within an orifice of the native mitral valve; wherein the device (300; 400) is configured such that, when the device (300; 400) is implanted within the orifice by securing the sheet to the annulus of the native mitral valve above the first leaflet of the native valve: the free end portion (306; 412) is longer than the first leaflet; during diastole, the free end portion (306; 412) extends along an atrial surface of the first leaflet and downwardly through the orifice and into a left ventricle of the heart; during diastole, the free end portion (306; 412) extends below a lower end of the first leaflet; during systole, the free end portion (306; 412) moves laterally: away from the first leaflet such that a gap exists between the free end portion (306; 412) and the first leaflet; and toward a second leaflet of the native mitral valve for helping to seal the orifice and reduce heart valve regurgitation during systole; and during systole, the free end portion (306; 412) extends short of a lower end of the second leaflet; during diastole, the free end portion (306; 412) moves away from the second leaflet and collapses against the first leaflet to allow blood to flow from a left atrium of the heart to the left ventricle during diastole. a flexible sheet comprising blood-impermeable material and having an upper end portion (302; 410) and a free end portion (306; 412) extending away from the upper end portion (302; 406); means for securing the upper end portion (302; 406) of the sheet to an annulus of the native mitral valve above a first leaflet of the native mitral valve, such that the device (300; 400) is implanted within an orifice of the native mitral valve; wherein the device (300; 400) is configured such that, when the device (300; 400) is implanted within the orifice by securing the sheet to the annulus of the native mitral valve above the first leaflet of the native valve: the free end portion (306; 412) is longer than the first leaflet; during diastole, the free end portion (306; 412) extends along an atrial surface of the first leaflet and downwardly through the orifice and into a left ventricle of the heart; during diastole, the free end portion (306; 412) extends below a lower end of the first leaflet; during systole, the free end portion (306; 412) moves laterally: away from the first leaflet such that a gap exists between the free end portion (306; 412) and the first leaflet; and toward a second leaflet of the native mitral valve for helping to seal the orifice and reduce heart valve regurgitation during systole; and during systole, the free end portion (306; 412) extends short of a lower end of the second leaflet; during diastole, the free end portion (306; 412) moves away from the second leaflet and collapses against the first leaflet to allow blood to flow from a left atrium of the heart to the left ventricle during diastole. the free end portion (306; 412) is longer than the first leaflet; during diastole, the free end portion (306; 412) extends along an atrial surface of the first leaflet and downwardly through the orifice and into a left ventricle of the heart; during diastole, the free end portion (306; 412) extends below a lower end of the first leaflet; during systole, the free end portion (306; 412) moves laterally: away from the first leaflet such that a gap exists between the free end portion (306; 412) and the first leaflet; and toward a second leaflet of the native mitral valve for helping to seal the orifice and reduce heart valve regurgitation during systole; and away from the first leaflet such that a gap exists between the free end portion (306; 412) and the first leaflet; and toward a second leaflet of the native mitral valve for helping to seal the orifice and reduce heart valve regurgitation during systole; and during systole, the free end portion (306; 412) extends short of a lower end of the second leaflet; during diastole, the free end portion (306; 412) moves away from the second leaflet and collapses against the first leaflet to allow blood to flow from a left atrium of the heart to the left ventricle during diastole.