Description:
<jats:title>Abstract</jats:title><jats:p>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<jats:italic>G</jats:italic>acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle<jats:italic>E</jats:italic>of homogeneous spaces for a group bundle<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathscr {G}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math></jats:alternatives></jats:inline-formula>, all over a reduced Stein space<jats:italic>X</jats:italic>with compatible actions of a reductive complex group on<jats:italic>E</jats:italic>,<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathscr {G}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math></jats:alternatives></jats:inline-formula>, and<jats:italic>X</jats:italic>. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a<jats:italic>G</jats:italic>-manifold being<jats:italic>G</jats:italic>-Oka.</jats:p>