The present book is a new, substantially enlarged, version of a previous set of lecture notes on diophantine analysis, published first in 2008 and then in revised form a few years later by the Edizioni della Scuola Normale. The content mixes a number of rather classical results on diophantine equations and diophantine approximation the basic theory of heights and a few more recent results and applications. The exposition has been generally kept at an elementary and essentially self-contained level, focusing on some main ideas rather than finer technical results, which can be obtained by similar methods. In fact, the book is also addressed to readers outside the relevant fields with the hope that more expert readers also might find something relevant. The second edition contains substantial new material in the form of new sections, supplements, remarks, and exercises. The arguments for the exercises are developed in full by means of 'hints' which, in fact, are much more than scattered suggestions and practically contain complete details. Occasionally, the remarks and the exercises contain miscellaneous results which have not been explicitly published elsewhere. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
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