We already have the Milnor-Svarc Lemma, which tells us that if a group acts "nicely" on a space, then the Cayley graph of the group is quasi-isometric to the given space. This gives us a lot of hyperbolic groups, because hyperbolic manifolds are ubiquitous.
Edit: For example, the MS-lemma tells us that the surface groups are hyperbolic.
Of course, in the large-scale geometric setting here, we consider two metric spaces equivalent if they are quasi-isometric.
There is a result of Kharlampovich and Myasnikov, in "Hyperbolic groups and free constructions" which allows us to construct a "separated HNN extension" of two hyperbolic groups to yield a hyperbolic group.
What are some other ways of constructing word-hyperbolic groups?