Devin Balkcom has built a robot that folds origami. Well, very simple origami at least. It's fascinating to see the efforts of robots at emulating human manipulations and to see just how clumsy they are. I would have thought machines could fold paper pretty well by now, but maybe not in a generalized way.
It's also eye-opening to see the academic thought process at work:
One interesting result to think about — an ordinary shopping bag cannot be folded flat without bending the facets. (The flat and open configurations are isolated points.) This might be considered a 'design feature'; the fact that the facets resist bending tends to keep the shopping bag open or closed. The fact that paper is not stretchy suggests that it may even be the case that bending the facets is not enough, and that some crinkling of the paper must occur.