'Learning some topics in mathematics is a bit like climbing a mountain - it is best done with a guide. In this short course, Dundas is just that guide - revealing the best routes, giving the reader first-hand experience through lots of well-chosen exercises, providing relevant and motivating examples, and finally, making it all fun.' John McCleary, Vassar College, New York 'For such studies, the present book is excellent. It scores on a number of counts: It does a solid job on the big topics that launch the subject i.e., manifolds, the tangent space, the cotangent space - the usual suspects, as Claude Rains would have it (also of differential geometry) ... The book starts with some marvelous and - at least to me - unexpected motivations, to wit, a discussion of how a robot's arm operating in 3-space sweeps out surprising manifolds (like the torus), a discussion of the configuration space of a pair of electrons, and a discussion of state spaces and fibre bundles ... You gotta love it. It looks like a very good book.' Michael Berg, MAA Reviews '... the author has presented a wealth of important basic material of manifold theory ... in a highly enlightening, inspiring and lucid manner. No doubt, this short introduction to the subject is even more than that, namely also a gripping guide through a fascinating area of contemporary mathematics. As such it may serve as an excellent introduction to the many more advanced textbooks in the literature on differential topology.' Werner Kleinert, zbMATH 'Summing up, the book is highly recommendable for all publics. An interested student can very well go through it quite by her/himself and learn a lot. A professor surely can discover useful aspects that may have skipped her/his attention before. But most important, Professor Dundas offers us a very enjoyable reading!' Jesus M. Ruiz, European Mathematical Society (www.euro-math-soc.eu) '... the author writes in a style that students will enjoy reading: the book is quite clear, chatty to the point of being colloquial ... and filled with helpful examples ... [It] serves beautifully as a first look at this material, and does a splendid job of preparing the reader for more sophisticated accounts of the subject.' Mark Hunacek, The Mathematical Gazette