Devices and methods for acquiring and fastening tissues folds within an internal organ, such as the stomach, and for applying the methods and devices to producing reductions in organ volume or repair of bariatric procedures, are disclosed. An exemplary method for forming a continuous laterally extending tissue fold involves forming a succession of laterally extending folds having adjacent overlapping fold portions. One exemplary tissue-acquisition device has an open-end roller-and-arm structure that allows individual tissue folds to be formed and fastened, then advanced to an adjacent region within the stomach, for capture of a new fold that will form an extension of the existing fold(s).