Although research in curve shortening flow has been very active for ne arly 20 years, the results of those efforts have remained scattered th roughout the literature. For the first time, The Curve Shortening Prob lem collects and illuminates those results in a comprehensive, rigorou s, and self-contained account of the fundamental results. The authors present a complete treatment of the Gage-Hamilton theorem, a clear, de tailed exposition of Grayson's convexity theorem, a systematic discuss ion of invariant solutions, applications to the existence of simple cl osed geodesics on a surface, and...