I'd like to have random tessellations of regions in a hyperbolic space.
In the Euclidean plane I get good results by scattering random points and performing a periodic Delaunay triangulation using CGAL.
For the hyperbolic case, though, there is nothing yet available in the library, even tough work on the implementation of non-Euclidean triangulations and meshes in CGAL was ongoing already in 2011, and essentially ready by 2014.
A purportedly "easy" recipe for implementing the hyperbolic triangulation has been long available (arxiv.org:0903.3287), but I don't think it's trivial to implement it reliably.
Is there any other implementation of hyperbolic Delaunay triangulations, preferably with periodic boundary conditions?
The code that Marc mentions is computing periodic triangulations (along translations corresponding to the hyperbolic octagon), following the paper soon to be presented at SoCG'17 (see https://hal.inria.fr/hal-01411415 for a preliminary version).
We also have code that computes Delaunay triangulations in the hyperbolic plane, as presented in our JoCG paper (see http://jocg.org/index.php/jocg/article/view/141). The branch is currently private in github, but we will make it public soon. Some parts need polishing, though, and the documentation is not yet written.