  In this thesis, I am going to consider Bernoulli percolation on graphs
admitting vertex-transitive actions of groups of isometries of d-dimensional
hyperbolic spaces H^d. In the first chapter, I give an overview concerning
percolation and its terminology. In the separate introduction to each of the
other chapters, I explain its contents more precisely, giving also some
preliminaries needed therein. In the second chapter, I prove the existence of a
non-trivial non-uniqueness phase of Bernoulli percolation on Cayley graphs for
a wide class of Coxeter reflection groups of finite type polyhedra in H^3. In
the third chapter, I consider some geometric property of the clusters in
Bernoulli bond percolation in the non-uniqueness phase on a class of connected,
transitive, locally finite graphs in H^d, for any d.
