This volume is the offspring of a week-long workshop on Galois groups over Q and related topics, which was held at the Mathematical Sciences Research Institute during the week March 23-27, 1987. The organizing committee consisted of Kenneth Ribet (chairman), Yasutaka Ihara, and Jean-Pierre Serre. The conference focused on three principal themes: 1. Extensions of Q with finite simple Galois groups. 2. Galois actions on fundamental groups, nilpotent extensions of Q arising from Fermat curves, and the interplay between Gauss sums and cyclotomic units. 3. Representations of Gal(Q/Q) with values in GL(2)j deformations and connections with modular forms. Here is a summary of the conference program: * G. Anderson: Gauss sums, circular units and the simplex * G. Anderson and Y. Ihara: Galois actions on 11 1 ( *** ) and higher circular units * D. Blasius: Maass forms and Galois representations * P. Deligne: Galois action on 1I 1(P-{0, 1, oo}) and Hodge analogue * W. Feit: Some Galois groups over number fields * Y. Ihara: Arithmetic aspect of Galois actions on 1I 1(P - {O, 1, oo}) - survey talk * U. Jannsen: Galois cohomology of i-adic representations * B. Matzat: - Rationality criteria for Galois extensions - How to construct polynomials with Galois group Mll over Q * B. Mazur: Deforming GL(2) Galois representations * K. Ribet: Lowering the level of modular representations of Gal( Q/ Q) * J-P. Serre: - Introductory Lecture - Degree 2 modular representations of Gal(Q/Q) * J.