this book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. the methods are classical, and the results stressed can be obtained without much background in algebraic geometry. in particular, thue equations, norm form equations and s-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. the book will be useful for graduate students and researchers. (lenseignement mathematique) the rich bibliography includes more than hundred references. the book is easy to read, it may be a useful piece of reading not only for experts but for students as well. acta scientiarum mathematicarum