The purpose of this book is to provide an accessible introduction to a new set of methods for the analysis of Lagrangian motion in geophysical flows. These methods were originally developed in the abstract mathem- ical setting of dynamical systems theory, through a geometric approach to differential equations that ultimately owes much to the insights of Poincar´ e (1892). In the 1980s and 1990s, researchers in applied mathematics and—fluid dynamics recognized the potential of this approach for the analysis of fluid motion. Despite these developments and the existence of a substantial body of work on geophysical fluid problems in the dynamical systems and geophysicalliterature,nointroductorytexthasbeenavailablethatpresents these methods in the context of geophysical fluid flow. The text is meant to be accessible to geophysical fluid scientists and students familiar with the mathematics of ordinary (mostly) and partial (sometimes) differential equations. It assumes little or no prior knowledge of dynamical systems theory. An effort is made to explain concepts from a physical point of view, and to avoid the theorem and proof constructions that appear in dynamical systems texts. We hope that this book will prove usefultograduatestudents,researchscientists,andeducatorsinanybranch of geophysical fluid science in which the motion and transport of fluid, and ofmaterialscarriedbythefluid,isofinterest.Wehopethatitwillalsoprove interesting and useful to applied mathematicians who seek an introduction to an intriguing and rapidly developing area of geophysical fluid dynamics.