Surgical implants use combined anchoring components to replace meniscal or labral cartilage, in ways that provide strong reinforcement while emulating natural anchoring. An arc-shaped polymer segment is coupled to an anchoring rim made of shape-memory material, which will fit into a groove prepared in a bone surface using specialized tools. A fabric material or anchoring ring is provided above the polymer segment, and can be secured to a knee capsule or other soft tissue. Fabric strips can extend out from the tips of the polymer arc, for additional anchoring. An additional polymer segment can also be provided to replace a hyaline cartilage layer, with a porous bottom surface to promote tissue ingrowth. By using peripheral rather than central anchoring, such implants can be given very high strength and stbility, to last for multiple decades.