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...