  A basic problem in the study of algebraic morphisms is to determine which
sets can be realised as the image of an endomorphism of affine space. This
paper extends the results previously obtained by the first author on the
question of existence of surjective maps $F\colon \mathbb{A}^n \rightarrow
\mathbb{A}^n\setminus Z$, where $Z$ is an algebraic subvariety of
$\mathbb{A}^n$ of codimension at least 2. In particular, we show that for any
(affine) algebraic variety $Z$ of dimension at most $n-2$, there is an
algebraic variety $W\subset \mathbb{A}^n$ birational to $Z$ and a surjective
algebraic morphism $\mathbb{A}^n\rightarrow \mathbb{A}^n\setminus W$. We also
propose a conjectural approach towards resolving unknown cases.
