Geometric dissection is the mathematical art of cutting figures into pieces that can be rearranged to form other figures, using as few pieces as possible. Each dissection problem can be viewed as a puzzle, and many, such as the dissection of a regular octagon to a square, have elegant solutions. I will examine a selection of remarkable dissections, employing a variety of solution methods, including strips, steps, tessellations, and an exploration of star and polygon structures. In addition, I will comment on the colorful history of these problems, which originated many centuries ago and have flourished ever since their appearance in the mathematical puzzle columns of turn-of-the-century newspapers. The talk is based on my recent book, "Dissections: Plane & Fancy", Cambridge University Press, 1997.