Given a smooth surface in R^3, a classical question in differential geometry asks whether the surface can be continuously deformed through a smooth, nontrivial family of isometric surfaces. If such a family exists and does not arise from rigid motions of R^3, then the surface is said to be flexible. An old conjecture asserts that flexible, smooth closed surfaces do not exist. In this talk, we survey this question and the general uniqueness problem for isometric immersions. We then present new examples of flexible, smooth immersed cylinders in R^3 which are neither minimal nor developable. We conclude with a discussion of speculative approaches to the construction of flexible, smooth closed surfaces. These results are part of upcoming work with Andrew Sageman-Furnas.