This volume explores Diophantine approximation on smooth manifolds embedded in Euclidean space, developing a coherent body of theory comparable to that of classical Diophantine approximation. In particular, the book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. After setting out the necessary background material, the authors give a full discussion of Hausdorff dimension and its uses in Diophantine approximation. They employ a wide range of techniques from the number theory arsenal to obtain the upper and lower bounds required, highlighting the difficulty of some of the questions considered. The authors then go on to consider briefly the p-adic case, and conclude with a chapter on some applications of metric Diophantine approximation. All researchers with an interest in Diophantine approximation will want to have this book in their personal libraries.