  The differential calculus based on the derivations of an associative algebra
underlies most of the noncommutative field theories considered so far. We
review the essential properties of this framework and the main features of
noncommutative connections in the case of non graded associative unital
algebras with involution. We extend this framework to the case of
${\mathbb{Z}}_2$-graded unital involutive algebras. We show, in the case of the
Moyal algebra or some related ${\mathbb{Z}}_2$-graded version of it, that the
derivation based differential calculus is a suitable framework to construct
Yang-Mills-Higgs type models on Moyal (or related) algebras, the covariant
coordinates having in particular a natural interpretation as Higgs fields. We
also exhibit, in one situation, a link between the renormalisable NC
$\phi^4$-model with harmonic term and a gauge theory model. Some possible
consequences of this are briefly discussed.
