Compliant shell mechanisms are creased and corrugated thin-walled structures that can drastically change shape to move, deploy, or adapt to a changing environment. They have found use cases in the context of recent space programs and in other domains ranging from biomedical technology to architecture. Not unlike slender beams, thin shells prefer bending over stretching. Ideally, thin shells deform isometrically should isometric deformations exist. The problem of finding, or disproving the existence of, isometric deformations for various surfaces preoccupied many mathematicians and mechanicians. The most noteworthy results undoubtedly pertain to three broad categories of surfaces: developable surfaces, convex surfaces, and axisymmetric surfaces. In the modern context of computer graphics, discrete differential geometry and “Origami science,” more focus has been directed towards tri- and quad-based polyhedral surfaces. In this lecture, we report on recent results that characterize the isometric deformations of periodic surfaces, be them smooth or piecewise smooth, with straight or curved creases.