A prosthetic aortic heart valve for implantation at an implantation site having an annulus, the prosthetic aortic heart valve comprising: a radially expandable and compressible support frame comprising a plurality of strut members interconnected to each other to form a mesh structure comprising an inflow end and an outflow end, the mesh structure comprising a distended intermediate portion having a first diameter (D 1) at a first location, the intermediate portion tapering in a direction toward the inflow end to form an inflow end portion having a second, smaller diameter (D 2) at a second location, and tapering in a direction toward the outflow end to form an outflow end portion having a fourth diameter (D 4) that is less than the first diameter (D 1); and a leaflet assembly comprising plural leaflets (34a-34c) having respective inflow end portions and outflow end portions, the inflow end portions of the leaflets (34a-34c) being secured to the inside of the mesh structure at the inflow end portion of the mesh structure, and the outflow end portions of the leaflets (34a-34c) defining angularly spaced commissures, wherein the inflow end portion of the mesh structure has a flared portion that increases in diameter from the second location to a third location at the inflow end of the mesh structure having a third diameter (D 3), the first and third diameters (D 1, D 3) being greater than the second diameter (D 2), wherein the mesh structure has a curved shape along its entire length that gradually increases in diameter from the outflow end portion to the intermediate section, then gradually decreases in diameter from the intermediate section to a location on the inflow end portion, and then gradually increases in diameter to form the flared portion terminating at the inflow end, and wherein the support frame comprises a plurality of angularly spaced retaining arms that extend from the outflow end of the mesh structure and are configured to releasably connect to a valve-retaining mechanism of a delivery apparatus. a radially expandable and compressible support frame comprising a plurality of strut members interconnected to each other to form a mesh structure comprising an inflow end and an outflow end, the mesh structure comprising a distended intermediate portion having a first diameter (D 1) at a first location, the intermediate portion tapering in a direction toward the inflow end to form an inflow end portion having a second, smaller diameter (D 2) at a second location, and tapering in a direction toward the outflow end to form an outflow end portion having a fourth diameter (D 4) that is less than the first diameter (D 1); and a leaflet assembly comprising plural leaflets (34a-34c) having respective inflow end portions and outflow end portions, the inflow end portions of the leaflets (34a-34c) being secured to the inside of the mesh structure at the inflow end portion of the mesh structure, and the outflow end portions of the leaflets (34a-34c) defining angularly spaced commissures, wherein the inflow end portion of the mesh structure has a flared portion that increases in diameter from the second location to a third location at the inflow end of the mesh structure having a third diameter (D 3), the first and third diameters (D 1, D 3) being greater than the second diameter (D 2), wherein the mesh structure has a curved shape along its entire length that gradually increases in diameter from the outflow end portion to the intermediate section, then gradually decreases in diameter from the intermediate section to a location on the inflow end portion, and then gradually increases in diameter to form the flared portion terminating at the inflow end, and wherein the support frame comprises a plurality of angularly spaced retaining arms that extend from the outflow end of the mesh structure and are configured to releasably connect to a valve-retaining mechanism of a delivery apparatus.