  In this thesis noncommutative gauge theory is extended beyond the canonical
case, i.e. to structures where the commutator no longer is a constant. In the
first part noncommutative spaces created by star-products are studied. We are
able to identify differential operators that still have an undeformed Leibniz
rule and can therefore be gauged much in the same way as in the canonical case.
By linking these derivations to frames (vielbeins) of a curved manifold, it is
possible to formulate noncommutative gauge theories that admit nonconstant
noncommutativity and go to gauge theory on curved spacetime in the commutative
limit. We are also able to express the dependence of the noncommutative
quantities on their corresponding commutative counterparts by using
Seiberg-Witten maps. In the second part we study noncommutative gauge theory in
the matrix theory approach. There, the noncommutative space is a finite
dimensional matrix algebra (fuzzy space) which emerges as the ground state of a
matrix action, the fluctuations around this ground state creating the gauge
theory. This gauge theory is finite, goes to gauge theory on a 4-dimensional
manifold in the commutative limit and can also be used to regularize the
noncommutative gauge theory of the canonical case. In particular, we are able
to match parts of the known instanton sector of the canonical case with the
instantons of the finite theory.
