  We develop a formalism to realize algebras defined by relations on function
spaces. For this porpose we construct the Weyl-ordered star-product and present
a method how to calculate star-products with the help of commuting vector
fields. Concepts developed in noncommutative differential geometry will be
applied to this type of algebras and we construct actions for noncommutative
field theories. Derivations of star-products makes it further possible to
extend noncommutative gauge theory in the Seiberg-Witten formalism with
covariant derivatives. In the commutative limit these theories are becoming
gauge theories on curved backgrounds. We study observables of noncommutative
gauge theories and extend the concept of so called open Wilson lines to general
noncommutative gauge theories.
