  I recall the main motivation to study quantum field theories on
noncommutative spaces and comment on the most-studied example, the
noncommutative R^4. That algebra is given by the *-product which can be written
in (at least) two ways: in an integral form or an exponential form. These two
forms of the *-product are adapted to different classes of functions, which,
when using them to formulate field theory, lead to two versions of quantum
field theories on noncommutative R^4. The integral form requires functions of
rapid decay and a (preferably smooth) cut-off in the path integral, which
therefore should be evaluated by exact renormalisation group methods. The
exponential form is adapted to analytic functions with arbitrary behaviour at
infinity, so that Feynman graphs can be used to compute the path integral
(without cut-off) perturbatively.
