  We survey recent work, published since 2015, on equivariant Oka theory. The
main results described in the survey are as follows. Homotopy principles for
equivariant isomorphisms of Stein manifolds on which a reductive complex Lie
group $G$ acts. Applications to the linearisation problem. A parametric Oka
principle for sections of a bundle $E$ of homogeneous spaces for a group bundle
$\mathscr G$, all over a reduced Stein space $X$ with compatible actions of a
reductive complex group on $E$, $\mathscr G$, and $X$. Application to the
classification of generalised principal bundles with a group action. Finally,
an equivariant version of Gromov's Oka principle based on a new notion of a
$G$-manifold being $G$-Oka.
