A prosthetic heart valve (800B, 800C, 900A-D) configured to be expanded between leaflets of a native aortic valve of a patient, the prosthetic heart valve (800B, 800C, 900A-D) comprising: a collapsible and expandable stent (806, 906) extending in a flow direction between a proximal end (802, 902) and a distal end, the stent (806, 906) including an annulus section adjacent the proximal end (802, 902) and a plurality of struts (807, 907) shaped to form a plurality of cells (815, 915) connected to one another in a plurality of annular rows (813, 912) around the stent (806, 906), the flow direction being the direction from the proximal end toward the distal end; a cuff (812, 912) attached to the annulus section of the stent and having an outer surface facing in a radial direction orthogonal to the flow direction; a plurality of prosthetic valve leaflets (810, 910) attached to the cuff (812, 912); and a plurality of sealing members (820B, 820C, 825B, 825C, 920A-D, 925A-D) each attached to the annulus section of the cuff,each sealing member (820B, 820C, 825B, 825C, 920A-D, 925A-D) having an open side facing generally toward the distal end of the stent and a closed side facing generally toward the proximal end of the stent (806, 906), so that blood flowing in a direction opposite the flow direction will enter at least one of the sealing members (820B, 820C, 825B, 825C, 920A-D, 925A-D) through the open side, causing an outer surface of the at least one sealing member (820B, 820C, 825B, 825 C, 920A-D, 925A-D) to move away from the outer surface of the cuff (812, 912) wherein the sealing members (820B, 820C, 825B, 825C, 920A-D, 925A-D) take the form of parachute-like elements billowing open when blood flows in a direction opposite the flow direction into the parachute-like elements, characterized in that at least one of the sealing members (825B, 825 C, 925A-D) extends across two or three cells in a circumferential direction orthogonal to the flow direction and the radial direction. a collapsible and expandable stent (806, 906) extending in a flow direction between a proximal end (802, 902) and a distal end, the stent (806, 906) including an annulus section adjacent the proximal end (802, 902) and a plurality of struts (807, 907) shaped to form a plurality of cells (815, 915) connected to one another in a plurality of annular rows (813, 912) around the stent (806, 906), the flow direction being the direction from the proximal end toward the distal end; a cuff (812, 912) attached to the annulus section of the stent and having an outer surface facing in a radial direction orthogonal to the flow direction; a plurality of prosthetic valve leaflets (810, 910) attached to the cuff (812, 912); and a plurality of sealing members (820B, 820C, 825B, 825C, 920A-D, 925A-D) each attached to the annulus section of the cuff,each sealing member (820B, 820C, 825B, 825C, 920A-D, 925A-D) having an open side facing generally toward the distal end of the stent and a closed side facing generally toward the proximal end of the stent (806, 906), so that blood flowing in a direction opposite the flow direction will enter at least one of the sealing members (820B, 820C, 825B, 825C, 920A-D, 925A-D) through the open side, causing an outer surface of the at least one sealing member (820B, 820C, 825B, 825 C, 920A-D, 925A-D) to move away from the outer surface of the cuff (812, 912) wherein the sealing members (820B, 820C, 825B, 825C, 920A-D, 925A-D) take the form of parachute-like elements billowing open when blood flows in a direction opposite the flow direction into the parachute-like elements,