In principle, a torus is obtained by gluing the opposite edges of a rectangle to each other (with the correct orientations). So, what about in practice? Is it possible to literally bend/fold/roll a flat piece of paper and tape the opposite edges together? As it relates to the torus, this question has been explored on a number of fronts, so we ask: what about other surfaces? What would it take to realize an origami Klein bottle, two-holed torus, or projective plane? How faithfully can we respect both the topology and geometry of constant curvature surfaces in euclidean three-space? Provisionally labelling this cloud of questions as the study of "topological origami", I'll share some examples and illustrations and suggest potential next steps.