  We establish a precise relation between Galois theory in its motivic form
with the mathematical theory of perturbative renormalization (in the minimal
subtraction scheme with dimensional regularization). We identify, through a
Riemann-Hilbert correspondence based on the Birkhoff decomposition and the
t'Hooft relations, a universal symmetry group (the "cosmic Galois group"
suggested by Cartier), which contains the renormalization group and acts on the
set of physical theories. This group is closely related to motivic Galois
theory. We construct a universal singular frame of geometric nature, in which
all divergences disappear. The paper includes a detailed overview of the work
of Connes-Kreimer and background material on the main quantum field theoretic
and algebro-geometric notions involved. We give a complete account of our
results announced in math.NT/0409306.
