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

24 pages, An extended abstract of this paper will appear in the proceedings of CIAC 2015