Folding and coloring problems in mathematics and physics

We review various folding problems arising in the physics of membranes and polymers. These are (1) the phantom folding of tethered membranes, i.e. the two-dimensional lattice folding; (2) the phantom folding of fluid membrane s, i.e. the folding of tessellations of arbitrary genus; (3) the self-avoid ing folding of polymers, i.e. the meander problem. All three problems are f ound to be related to coloring problems and possess one kind of underlying integrable structure, in different guises. Many mathematical results follow from taking advantage of this fact.