Transport barriers offer a simplified global template for the redistribution ofsubstances without the need to simulate or observe numerous different initial distributions in detail. Because of their simplifying role, transport barriers are broadly invoked as explanations for observations in several physical disciplines, including geophysical flows,fluid dynamics,plasma fusion, reactive flowsand molecular dynamics. Despite their frequent conceptual use, however, transport barriers are rarely defined precisely or extracted systematically from data. The purpose of this book is to survey effective and mathematically grounded methods for defining, locating and leveraging transport barriers in numerical simulations, laboratory experiments, technological processes and nature. In the rest of this Introduction, we briefly survey the main topics that we will be covering in later chapters.