  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.
