Is a smooth closed surface in Euclidean 3-space rigid? Classical theorem of Cohn-Vossen: A closed convex surface in Euclidean 3-space cannot be deformed isometrically.
Robert Connelly found an example of a polyhedral surface that can be deformed isometrically. A metal hinged model of it can be found at IHES.
But what about an arbitrary not-necessarily-convex smooth closed surface? Is it necessarily rigid? Or maybe it might be possible to make a smooth version of Connelly's example? It's easy to make smooth "hinges". The real challenge is finding a smooth model of the vertices, which is where two or more hinges meet.
 A: On the Springer Online Encyclopedia there's a relevant article here:
Tight and taut immersions.
It says that a theorem due to Kuiper in 1955 implies that no smooth closed surface in $R^3$ is $C^1$-isometrically rigid. I think the reference is: N.H. Kuiper,  On $C^1$-isometric embeddings, Indag. Math. XVII, (1955) 545–556 and 683–689. MR0075640, Zbl 0067.39601.
On the other hand it says that nothing is known for the $C^2$ case, and a book on Open Problems in Geometry also says that as 1994 it is still open.
As for polyhedra, Schlenker has a rigidity criterion for non-convex ones, preprint here: A rigidity criterion for non-convex polyhedra (sadly lacks the drawings, the published reference is Discrete and Computational Geometry, 33 (2005):2, 207–221. MR2121293, Zbl 1083.52006).
(I'm no expert on this, just some googling).
A: Apparently the question is still open for smooth enough surfaces and deformations (that is, at least $C^2$).
Mike Anderson wrote a preprint claiming to prove local rigidity of smooth enough surfaces, but it was later withdrawn.
Idjad Sabitov and his collaborators have been working on this question, developing for instance a theory of higher-order isometric deformations, see e.g. Sabitov, I. Kh. Local theory of bendings of surfaces [MR1039820 (91c:53004)]. Geometry, III, 179–256, Encyclopaedia Math. Sci., 48, Springer, Berlin, 1992. He conjectures that local rigidity holds for analytic surfaces.
