Here I'm maintaining a list of articles about origami-math that I think are particularly interesting and that are available on-line. I especially like survey articles, since they offer a good way to be introduced to the subject.

• A Mathematical Theory of Origami Constructions and Numbers by Roger C. Alperin, New York Journal of Mathematics, Vol. 6 (2000), 119-133.
  While not the first analysis of origami constructible numbers, it's a pretty good one. Alperin looks at what set of contructible numbers are generated by various subsets of the origami "axioms." He also presents various aspects of this material in a context of projective geometry, which I find particularly interesting.

• Sul metodo del ripiegamento della carta per la risoluzione dei problemi geometrici (in Italian) by Margherita P. Beloch, Periodico di Mathematiche, Ser. 4, Vol. 16, 1936, 104-108.
  This is, perhaps, the first published paper containing a "deep" result of origami geometry. In it, Beloch describes how she discovered how paper folding can solve general cubic equations while teaching a class in 1933 at the University of Ferrara, Italy. Beloch was an accomplished algebraic geometer of her day, and it is unfortunate how her work on paper folding has gone largely unnoticed (or unremembered) by many recent writers on the subject. (I, too, made this mistake in a 1996 Monthly paper I wrote.)
  Professor Giorgio Ferrarese of the University of Turin has placed scans of Beloch's paper folding articles on the web, as well as many other articles having to do with Lill's method and such.

• Recent Results in Computational Origami by Erik D. Demaine and Martin L. Demaine, Origami³: Third International Meeting of Origami Science, Mathematics, and Education, T. Hull ed., AK Peters (Natick, MA), 2002, 3-16.
  This is a great survey article on the field of computational origami. It probably is a bit out-dated now, but it still makes for a fine introduction to the subject. More information on this topic can, of course, be found in the book that Erik Demaine and Joe O'Rourke wrote, Geometric Folding Algorithms (Campridge University Press, 2007).

• The combinatorics of flat folds: a survey by Thomas C. Hull, Origami³: Third International Meeting of Origami Science, Mathematics, and Education, T. Hull, ed., AK Peters (Natick, MA), 2002, 29-38.
  It is pretty sketchy for me to include one of my own papers here, but right now this one seems to be the only survey paper out there on combinatorial modeling of paper folding. If you're interested in things like Kawasaki's Theorem, Maekawa's Theorem, and determining how many valid ways you can assign mountains and valleys to the creases of a flat origami model, check this paper out.

• Resolution par le pliage de l'equation du troisieme degre et applications geometriques (in French), by Jacques Justin, L'Ouvert: Journal of the APMEP of Alsace and the IREM of Strasbourg, No. 42, March 1986, 9-19.
  To my knowledge, this is the first published article that compiles a list of origami "axioms", or basic moves, that define origami geometric constructions. Justin's list of basic moves contains all the ones that Scimemi, Huzita, Geretschlaeger, and Hatori independently developed later. In this paper Justin mentioned that his list was "inspired" by a list made by Peter Messer, but Messer's list was never actually published. Rather, it was privately circulated as a hand-written, photocopied manusciript. Messer's list contains the same "axioms" that Scimemi and Huzita describe in their 1989 paper, but Justin adds the additional one that Hatori included in 2003.
  Lucky for us, the French magazine L'Ouvert keeps most of its articles archived online! Notice how at the end of the article Justin includes an addendum where he credits Beloch for being the first to discover, in the 1930s, the "2 points folded to 2 lines" move and prove that it can solve cubic equations. He further credits Huzita for making him aware of Beloch's work.