Graph pebbling is a game in which a collection of pebbles is placed on the vertices of a connected graph. A pebbling move consists of removing two pebbles from one vertex and placing a pebble on any adjacent vertex. Of interest is the pebbling number of a graph G: the least k such that for any collection of k pebbles on V(G) and any target v ∈ V(G), there exists a sequence of pebbling moves placing a pebble on v. It is well-known that the pebbling number of a graph G is at least |V(G)| and is, in general, bounded above by an exponential function of |V(G)|. The Erdős-Rényi random graph model has frequently been used to address questions relating to the ‘typical’ behaviour of a graph with regards to some interesting property. This model has been used to show that almost all graphs G with at least a constant fraction of the possible edges have pebbling number |V(G)|. In addition to a survey of selected pebbling results and techniques, this thesis presents a first- of-its-kind result showing that even amongst graphs with a much smaller proportion of edges, graphs of large pebbling number are rare: almost all graphs on n vertices and more than (nlogn)/2 edges have pebbling number bounded above by a linear function of n – significantly smaller than the exponential in n bound for arbitrary graphs. Furthermore, it is shown that almost all graphs on n vertices with more than (n/2)exp(√log n) edges have pebbling number n, an improvement over the current best result. Various additional original results are given, including a strongly improved upper bound on the pebbling number of the Kneser graph Kn(7,3).