Bugeaud (Université Louis Pasteur) surveys recent results on algebraic approximations and classifications. Starting from continued fractions and Khintchine's theorem, he introduces a variety of techniques, ranging from explicit constructions to metric number theory. The reader is led to advanced results such as the proof of Mahler's conjecture on S-numbers. Brief consideration is given to the p-adic and the formal power series cases. Some 40 exercises are included. The book can be used for a graduate course on Diophantine approximation, or as an introduction for non-experts. Specialists will appreciate the collection of 50 open problems.