  We survey results on the foldability of flat origami models. The main topics
are the question of when a given crease pattern can fold flat, the
combinatorics of mountain and valley creases, and counting how many ways a
given crease pattern can be folded. In particular, we explore generalizations
of Maekawa's and Kawasaki's Theorems, develop a necessary and sufficient
condition for a given assignment of mountains and valleys to fold up in a
special case of single vertex folds, and describe recursive formulas to
enumerate the number of ways that single vertex in a crease pattern can be
folded.
