  A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots
\}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list
assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in
V$, $L(u)$ is a subset of $\{1, 2, \dots\}$ of size $k$. A colouring $c$ of $G$
respects a $k$-regular list assignment $L$ of $G$ if $c(u)\in L(u)$ for every
$u\in V$. A graph $G$ is $k$-choosable if for every $k$-regular list assignment
$L$ of $G$, there exists a colouring of $G$ that respects $L$. We may also ask
if for a given $k$-regular list assignment $L$ of a given graph $G$, there
exists a colouring of $G$ that respects $L$. This yields the $k$-Regular List
Colouring problem. For $k\in \{3,4\}$ we determine a family of classes ${\cal
G}$ of planar graphs, such that either $k$-Regular List Colouring is
NP-complete for instances $(G,L)$ with $G\in {\cal G}$, or every $G\in {\cal
G}$ is $k$-choosable. By using known examples of non-$3$-choosable and
non-$4$-choosable graphs, this enables us to classify the complexity of
$k$-Regular List Colouring restricted to planar graphs, planar bipartite
graphs, planar triangle-free graphs and to planar graphs with no $4$-cycles and
no $5$-cycles. We also classify the complexity of $k$-Regular List Colouring
and a number of related colouring problems for graphs with bounded maximum
degree.
