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, and may be positioned/repositioned until proper alignment and positioning is achieved. 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 defining a first tissue engaging region and a second tissue engaging region spaced from the first tissue engaging region. The positioning elements are adapted to substantially completely invert by rotating relative to the resilient ring between a first position in which each of the first and second tissue engaging regions is inwardly directed for facilitating positioning of the valve prosthesis within a delivery catheter, and a second position in which each of the first and second tissue engaging regions is outwardly directed for engaging tissue. The valve prosthesis may also include a valve skirt mounted with respect to the resilient ring.