I will provide a sample of geometric folding algorithms in three areas, roughly one-dimensional (1D), 2D, and 3D. The folding of 1D linkages finds applications from robotics to protein folding. I will describe the recent resolution of a 25-yr old open problem, showing that a chain cannot lock in the plane, and connect this result to morphing in computer graphics. Folding 2D paper leads to questions in mathematical origami. Here I'll describe the one-cut theorem: any straight-line drawing may be cut out of a folded piece of paper via one scissors cut. Unfolding the surface of 3D polyhedra has application to manufacturing, where shapes are cut out of sheets of aluminum and folded by metal-bending machines into 3D. I will highlight a long-unsolved problem, and discuss the recent resolution of a special case, unfolding polyhedra whose faces meet at right angles.