Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (page 6, link below). Is there any information available about the current status of this conjecture?
As already mentioned in the comments, it is still unknown whether hyperbolic groups are CAT(-1) or even CAT(0). A related question is:
Let $G$ be a hyperbolic group (endowed with a finite generated set). If $d$ is sufficiently large, does there exist a $G$-equivariant CAT(-1) or CAT(0) metric defined on the Rips complex $P_d(G)$?
This question is also open, but a counterexample is given here (Corollary 5.10) for $P_d(X)$ where $X$ is a specific (locally infinite) quasi-tree.
An alternative approach is to replace Rips' complex with the injective hull of the group. See the last paragraph of the introduction of Lang's paper.
Actually, even the following question is open:
Let $G$ be a hyperbolic group acting geometrically on a CAT(0) cube complex. Does $G$ act geometrically on a CAT(-1) cube complex?
The question appears for instance here.
Also relevant is this article of Brady and Crips constructing a hyperbolic group with CAT(0) dimension two but CAT(-1) dimension three.