  We say that a vertex colouring $\varphi$ of a graph $G$ is nonrepetitive if
there is no positive integer $n$ and a path on $2n$ vertices $v_{1}\ldots
v_{2n}$ in $G$ such that the associated sequence of colours
$\varphi(v_{1})\ldots\varphi(v_{2n})$ satisfy $\varphi(v_{i})=\varphi(v_{i+n})$
for all $i=1,2,\dots,n$. The minimum number of colours in a nonrepetitive
vertex colouring of $G$ is the Thue chromatic number $\pi (G)$. For the case of
vertex list colourings the Thue choice number $\pi_{l}(G)$ of $G$ denotes the
smallest integer $k$ such that for every list assignment $L:V(G)\rightarrow
2^{\mathbb{N}}$ with minimum list length at least $k$, there is a nonrepetitive
vertex colouring of $G$ from the assigned lists. Recently it was proved that
the Thue chromatic number and the Thue choice number of the same graph may have
an arbitrary large difference in some classes of graphs. Here we give an
overview of the known results where we compare these two parameters for several
families of graphs and we also give a list of open problems on this topic.
