Absolute Galois groups of fields often contain vital information about fields. They form an important ingredient in the study of deep arithmetical questions and problems in algebraic geometry and related areas. Absolute Galois groups are profinite groups. A major difficult classical problem is the characterization of absolute Galois groups among profinite groups.

Although it seems that we are quite far from a complete solution of this problem, there has been some remarkable progress in learning about some properties of absolute Galois groups.

In this talk I will survey some results related to the small important quotients of absolute Galois groups, the Rost-Voevodsky theorem (formerly the Bloch-Kato conjecture), and the Massey vanishing conjecture