In this thesis, we study a recently proposed model of random graphs that exhibit properties which are present in a wide range of networks arising in real world settings. The model creates random geometric graphs on the hyperbolic plane, where vertices are connected if they are within a certain threshold distance. We study typical properties of these graphs. We identify two critical values for one of the parameters that act as sharp thresholds. The three resulting intervals of the parameters that correspond to three possible phases of the random structure: A.a.s., the graph is connected; A.a.s., the graph is not connected, yet there is a giant component; A.a.s., every component is of sublinear size. Furthermore, we determine the behaviour at the critical values. We also consider typical distances between vertices and show that the ultra-small world phenomenon is present. Our results imply that most pairs of vertices that belong to the giant component are within doubly logarithmic distance.