In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.

Contributions by: F. Amoroso, D. Bertrand, W.D. Brownawell, G. Diaz, M. Laurent, Yu.V. Nesterenko, K. Nishioka, P. Philippon, G. Remond, D. Roy, M. Waldschmidt