Valve prostheses are disclosed that are adapted for secure and aligned placement relative to a heart annulus. The valve prostheses may be placed in a non-invasive manner, e.g., via trans-catheter techniques. The valve prosthesis may include a resilient ring, a plurality of leaflet membranes mounted with respect to the resilient ring, and a plurality of positioning elements movably mounted with respect to the flexible ring. Each of the positioning elements defines respective proximal, intermediate, and distal tissue engaging regions cooperatively configured and dimensioned to simultaneously engage separate corresponding areas of the tissue of an anatomical structure, including respective first, second, and third elongate tissue-piercing elements. The proximal, distal, and intermediate tissue-engaging regions are cooperatively configured and dimensioned to simultaneously engage separate corresponding areas of the tissue of an anatomical structure so as to stabilize a position of the valve prosthesis with respect to the anatomical structure, including wherein for purposes of so simultaneously engaging the separate corresponding areas of tissue, at least one of the first, second, and third elongate tissue-piercing elements is pointed at least partially opposite the direction of blood flow, and at least another thereof is pointed at least partially along the direction of blood flow. The valve prosthesis may also include a skirt mounted with respect to the resilient ring for sealing a periphery of the valve prosthesis against a reverse flow of blood around the valve prosthesis.