  We consider the groups $\operatorname{Diff}_{\mathcal B}(\mathbb R^n)$,
$\operatorname{Diff}_{H^\infty}(\mathbb R^n)$, and
$\operatorname{Diff}_{\mathcal S}(\mathbb R^n)$ of smooth diffeomorphisms on
$\mathbb R^n$ which differ from the identity by a function which is in either
$\mathcal B$ (bounded in all derivatives), $H^\infty = \bigcap_{k\ge 0}H^k$, or
$\mathcal S$ (rapidly decreasing). We show that all these groups are smooth
regular Lie groups.
