Apparatus and methods for occluding hollow body structures, such as blood vessels, and for attaching tissue layers together by providing implantable elements on opposite sides of the structure or tissue layers and drawing the implants together to occlude the body structure and/or bring the tissue layers together. The implants are deliverable in a low-profile configuration and self-expand to an enlarged configuration. The implantable elements are delivered by transfixing the body structure, then releasing the implants on opposite sides of the body structure and drawing the implants together to effect an occlusion or attachment. The implants are configured to apply oppositely directed forces to opposite surfaces of the tissue layers at alternate, circumferentially spaced locations and may constrain the tissue in a serpentine pattern or in a direct clamping pattern. The implants grip the tissue in a manner that defines a pressure zone about the transfixion aperture that prevents leakage from the aperture. The implants have a low profile in that they have a relatively short axial dimension relative to their deployed diameter.