Methods and apparatus for passing attachment apparatus (e.g., connector devices, staples, etc.) or connector material (e.g., suture thread, wire, cord, filament, monofilament, etc.) into or through the wall of a luminal anatomical structure (e.g., a blood vessel or other anatomical conduit) for the purpose of; i) closing the lumen of the anatomical structure, ii) forming an anastomotic junction between separate anatomical structures (or between approximated segments of the same anatomical structure), and/or iii) attaching an article (e.g., an endoluminal, extraluminal or transluminal graft) or other apparatus to the wall of the anatomical structure.