  A basic question of Diophantine approximation, which is the first issue we
discuss, is to investigate the rational approximations to a single real number.
Next, we consider the algebraic or polynomial approximations to a single
complex number, as well as the simultaneous approximation of powers of a real
number by rational numbers with the same denominator. Finally we study
generalisations of these questions to higher dimensions. Several recent
advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent,
T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these
works.
