  This paper gives a complete selfcontained proof of our result announced in
hep-th/9909126 showing that renormalization in quantum field theory is a
special instance of a general mathematical procedure of extraction of finite
values based on the Riemann-Hilbert problem.
  We shall first show that for any quantum field theory, the combinatorics of
Feynman graphs gives rise to a Hopf algebra $\Hc$ which is commutative as an
algebra. It is the dual Hopf algebra of the envelopping algebra of a Lie
algebra $\ud G$ whose basis is labelled by the one particle irreducible Feynman
graphs. The Lie bracket of two such graphs is computed from insertions of one
graph in the other and vice versa. The corresponding Lie group $G$ is the group
of characters of $\Hc$.
  We shall then show that, using dimensional regularization, the bare
(unrenormalized) theory gives rise to a loop $$ \g (z) \in G \qquad z \in C $$
where $C$ is a small circle of complex dimensions around the integer dimension
$D$ of space-time. Our main result is that the renormalized theory is just the
evaluation at $z = D$ of the holomorphic part $\g_+$ of the Birkhoff
decomposition of $\g$.
  We begin to analyse the group $G$ and show that it is a semi-direct product
of an easily understood abelian group by a highly non-trivial group closely
tied up with groups of diffeomorphisms. The analysis of this latter group as
well as the interpretation of the renormalization group and of anomalous
dimensions are the content of our second paper with the same overall title.
