The main theme of this book is a study of the embedding of the Galois group of an extension of the field K to the Galois group of a different extension satisfying appropriate conditions. In the simple cyclic case, it is possible to solve the problem in an elementary way. In general, it needs quite advanced methods of algebraic number theory. The book concentrates on embedding problems of the Brauer type and starts with Galois theory. Chapter 2 introduces obstructions as elements in a suitable cohomology group. The next three chapters review the main tools: Brauer groups, group cohomology and quadratic forms. Decompositions of obstructions are studied in chapter 6. The last two chapters contain a discussion of solvability of certain embedding problems and their possible reduction. An introduction to pro-finite Galois theory is contained in the appendix. The book will be useful as a reference book or as an introduction to the field.