  The concept of covariant coordinates on noncommutative spaces leads directly
to gauge theories with generalized noncommutative gauge fields of the type that
arises in string theory with background B-fields. The theory is naturally
expressed in terms of cochains in an appropriate cohomology; we discuss how it
fits into the framework of projective modules. The equivalence of star products
that arise from the background field with and without fluctuations and
Kontsevich's formality theorem allow an explicitly construction of a map that
relates ordinary gauge theory and noncommutative gauge theory (Seiberg-Witten
map.) As application we show the exact equality of the Dirac-Born-Infeld action
with B-field in the commutative setting and its semi-noncommutative cousin in
the intermediate picture. Using noncommutative extra dimensions the
construction is extended to noncommutative nonabelian gauge theory for
arbitrary gauge groups; an explicit map between abelian and nonabelian gauge
fields is given. All constructions are also valid for non-constant B-field,
Poisson structure and metric.
