Geometrically constrained statistical systems on regular and random lattices: From folding to meanders

We review a number a recent advances in the study of two-dimensional statistical models with strong geometrical constraints. These include folding problems of regular and random lattices as well as the famous meander problem of enumerating the topologically inequivalent configurations of a meandering road crossing a straight river through a given number of bridges. All these problems turn out to have reformulations in terms of fully packed loop models allowing for a unified Coulomb gas description of their statistical properties. A number of exact results and physically motivated conjectures are presented in detail, including the remarkable meander configuration exponent alpha=(29+sqrt(145))/12.