  We present a detailed study of the combinatorial interpretation of matrix
integrals, including the examples of tessellations of arbitrary genera, and
loop models on random surfaces. After reviewing their methods of solution, we
apply these to the study of various folding problems arising from physics,
including: the meander (or polymer folding) problem ``enumeration of all
topologically inequivalent closed non-intersecting plane curves intersecting a
line through a given number of points" and a fluid membrane folding problem
reformulated as that of ``enumerating all vertex-tricolored triangulations of
arbitrary genus, with given numbers of vertices of either color".
