The material in this booklet is concerned with a discussion and examination of geometric puzzles and the ideas which result from their study. The general idea of graphs is introduced as a tool which can be used to solve geometric puzzles. The fact that working with puzzles can lead to unexpected mathematical discoveries is stressed. Such topics as arcs and paths, Euler and Hamilton lines, dual graphs, and the Jordan Curve Theorem are discussed. Several puzzles are included which will provide many hours of entertainment for the interested reader. (FL)