The linearization problem asks if an action of a reductive algebraic group on complex affine $n$-space
$A^n$ is equivalent to a linear representation. For $A^2$ this is indeed the case,
due to the structure of the automorphism group of $A^2$ as an amalgamated product.
However, it does not hold in dimension $>2$.
The first counterexamples were given by G. W. Schwarz in 1989;
they initiated an interesting development.
A related object in this setting are affine space bundles, i.e. morphisms with all fibers
isomorphic to affine $n$-space. Here the fundamental question is if such a morphism is locally trivial
in some reasonable sense. We will describe some highlights, some open problems and some recent developments.