  We consider smooth families of Lie groups (group bundles) and connections
that are compatible with the group operation. We characterize the space of
group connections on a group bundle as an affine space modeled over the vector
space of $1$-forms with values cocycles in the Lie algebra bundle of the
aforementioned group bundle. We show that group connections satisfy the
Ambrose-Singer theorem and that group bundles can be seen as a particular case
of associated bundles realizing group connections as associated connections. We
give a construction of the Moduli space of group connections with fixed base
and fiber, as an space of representations of the fundamental group of the base.
