Geometric Puzzles: Fold and Cut

Fold a sheet of paper flat, make one complete straight cut, and unfold: what shapes can you get? I tour fold-and-cut puzzles, from a Japanese puzzle book of 1721 to the fold-and-cut theorem, which says that any shape with straight sides can be cut out this way. I describe the two known methods for finding the folds (one based on straight skeletons, one on disk packing), set warm-up puzzles with squares and triangles, and pose variations and a challenge to cut out a conference logo.