Embodiments of universal self-anchoring prosthetic valves for multi-position transcatheter or surgical implantation within any diseased or malfunctioning native heart valve are provided. An exemplary embodiment of a prosthetic valve includes a radially compressible and self-expanding central core housing a prosthetic valve, with compressible and flexible memory-shaped woven wire anchoring discs at the inflow and outflow ends. Specific design properties allow the prosthesis to be conformable, self-centering and flipped for multiple appropriate physiologic implant orientations. Expansive transverse radial force exerted by the central core, and memory-shape induced directional forces exerted by flexible inflow and outflow discs, capture and compress native peri-annular and leaflet tissues to anchor the prosthesis. Tissue facing frame surfaces may include small tines to enhance anchoring. The prosthetic valve frames may include various biomaterials or polymers as linings, coatings or coverings to enhance sealing. Methods and devices for delivering and implanting the valve based on access are described.