A consistent and near complete survey of the important progress made in the field over the last few years, with the main emphasis on the rigidity method and its applications. Among others, this monograph presents the most successful existence theorems known and construction methods for Galois extensions as well as solutions for embedding problems combined with a collection of the existing Galois realizations.

I. The Rigidity Method.- II. Applications of Rigidity.- III. Action of Braids.- IV. Embedding Problems.- V. Rigid Analytic Methods.