Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted f (G), is the minimal number of pebbles such that every configuration of f (G) pebbles on G is solvable. We derive several general upper bounds on the pebbling number, improving previous results.