  Recent results show that several important graph classes can be embedded as
subgraphs of strong products of simpler graphs classes (paths, small cliques,
or graphs of bounded treewidth). This paper develops general techniques to
bound the chromatic number (and its popular variants, such as fractional,
clustered, or defective chromatic number) of the strong product of general
graphs with simpler graphs classes, such as paths, and more generally graphs of
bounded treewidth. We also highlight important links between the study of
(fractional) clustered colouring of strong products and other topics, such as
asymptotic dimension in metric theory and topology, site percolation in
probability theory, and the Shannon capacity in information theory.
