Given a connected graph G and a distribution of non-negative integers on its vertices, a pebbling move on G is defined as the removal of two pebbles from one vertex, followed by the placement of one of those pebbles on an adjacent vertex. The pebbling number f(G) of a graph is the minimum number of pebbles needed such that, given any distribution of f(G) pebbles on G, one pebble can be placed on any specified but arbitrary vertex through a sequence of pebbling moves. It is known that for a graph G with n vertices, n ≤ f(G) ≤ 2^(n-1). In this talk, I will describe our attempts to determine which of the integers in the interval [n, 2^(n-1) ] can be realized as the pebbling number of a graph on n vertices.