This article is a slightly condensed reprint of Martin Gardner's very first "Mathematical Games" column (which appeared December 1956 in "Scientific American"). Here Gardner introduces flexagons (paper polygons folded from straight or crooked strips of paper which have the fascinating property of changing their faces when they are "flexed"), discusses their history prior to World War II, and explains how to make several hexaflexagons. The reprint, part of a special issue of "The College Mathematics Journal" devoted to "Martin Gardner's Mathematics," is followed by two contemporary papers that describe some of the ways that the study of flexagons has developed since 1956. Susan Goldstine and Ethan Berkhove provided vital assistance editing all three flexagon papers.