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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.