Mixing processes occur in a variety of technological and natural applications, with length and time scales ranging from the very small - as in microfluidic applications - to the very large - for example, mixing in the Earth's oceans and atmosphere. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification?The authors show how a range of flows in very different settings - micro to macro, fluids to solids - possess the characteristic of streamline crossing, a central kinematic feature of 'good mixing'. This notion can be placed on a firm mathematical footing via linked twist maps (LTMS), which are the central organizing principle of this book.The authors discuss the definition and construction of LTMS, provide examples of specific systems that can be analysed in the LTM framework and introduce a number of mathematical techniques - non-uniform hyperbolicity and smooth ergodic theory - which are then brought to bear on the problem of fluid mixing. In a final chapter, they argue that the analysis of linked twist maps opens the door to a plethora of a new investigations, both from the point of view of basic mathematics as well as new applications, and present a number of open problems and new directions. Consequently, this book will be of interest to a broad spectrum of readers, from pure and applied mathematicians, to engineers, physicists and geophysicists.