  Clique-width is a well-studied graph parameter owing to its use in
understanding algorithmic tractability: if the clique-width of a graph class
${\cal G}$ is bounded by a constant, a wide range of problems that are
NP-complete in general can be shown to be polynomial-time solvable on ${\cal
G}$. For this reason, the boundedness or unboundedness of clique-width has been
investigated and determined for many graph classes. We survey these results for
hereditary graph classes, which are the graph classes closed under taking
induced subgraphs. We then discuss the algorithmic consequences of these
results, in particular for the Colouring and Graph Isomorphism problems. We
also explain a possible strong connection between results on boundedness of
clique-width and on well-quasi-orderability by the induced subgraph relation
for hereditary graph classes.
