  We describe a set of methods to calculate gauge theory renormalization
constants from string theory, all based on a consistent prescription to
continue off shell open bosonic string amplitudes. We prove the consistency of
our prescription by explicitly evaluating the renormalizations of the two,
three and four-gluon amplitudes, and showing that they obey the appropriate
Ward identities. The field theory limit thus performed corresponds to the
background field method in Feynman gauge. We identify precisely the regions in
string moduli space that correspond to different classes of Feynman diagrams,
and in particular we show how to isolate contributions to the effective action.
Ultraviolet divergent terms are then encoded in a single string integral over
the modular parameter $\tau$. Finally, we derive a multiloop expression for the
effective action by computing the partition function of an open bosonic string
interacting with an external non-abelian background gauge field.
