  Noncommutative field theories on Moyal spaces can be conveniently handled
within a framework of noncommutative geometry. Several renormalisable matter
field theories that are now identified are briefly reviewed. The construction
of renormalisable gauge theories on these noncommutative Moyal spaces, which
remains so far a challenging problem, is then closely examined. The computation
in 4-D of the one-loop effective gauge theory generated from the integration
over a scalar field appearing in a renormalisable theory minimally coupled to
an external gauge potential is presented. The gauge invariant effective action
is found to involve, beyond the expected noncommutative version of the pure
Yang-Mills action, additional terms that may be interpreted as the gauge theory
counterpart of the harmonic term, which for the noncommutative $\phi^4$-theory
on Moyal space ensures renormalisability. A class of possible candidates for
renormalisable gauge theory actions defined on Moyal space is presented and
discussed.
