  A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for
each vertex, there is a list coloring, which is a coloring where each vertex
receives a color from its list. We study complexity issues of choosability of
graphs when the number $k$ of colors is limited. We get results which differ
surprisingly from the usual case where $k$ is implicit and which extend known
results for the usual case. We also exhibit some classes of graphs (defined by
structural properties of their blocks) which are choosable.
