We develop a new approach for the study of “typical” Riemann surfaces with high genus. The method that we use is the construction of random Riemann surfaces from oriented cubic graphs. This construction enables us to get a control over the global geometry properties of compact Riemann surfaces. We use the theory of random regular graphs to show that almost all such surfaces have large first eigenvalues and large Cheeger constants. Moreover a closer analysis of the probability space of oriented cubic graphs shows that on a typical surface there is a large embedded hyperbolic ball.