  If a graph has no induced subgraph isomorphic to any graph in a finite family
$\{H_1,\ldots,H_p\}$, it is said to be $(H_1,\ldots,H_p)$-free. The class of
$H$-free graphs has bounded clique-width if and only if $H$ is an induced
subgraph of the 4-vertex path $P_4$. We study the (un)boundedness of the
clique-width of graph classes defined by two forbidden induced subgraphs $H_1$
and $H_2$. Prior to our study it was not known whether the number of open cases
was finite. We provide a positive answer to this question. To reduce the number
of open cases we determine new graph classes of bounded clique-width and new
graph classes of unbounded clique-width. For obtaining the latter results we
first present a new, generic construction for graph classes of unbounded
clique-width. Our results settle the boundedness or unboundedness of the
clique-width of the class of $(H_1,H_2)$-free graphs (i) for all pairs
$(H_1,H_2)$, both of which are connected, except two non-equivalent cases, and
(ii) for all pairs $(H_1,H_2)$, at least one of which is not connected, except
11 non-equivalent cases.
  We also consider classes characterized by forbidding a finite family of
graphs $\{H_1,\ldots,H_p\}$ as subgraphs, minors and topological minors,
respectively, and completely determine which of these classes have bounded
clique-width. Finally, we show algorithmic consequences of our results for the
graph colour
