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We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of pebbles on the vertices. Player one wins if, through a series of pebbling moves, he can move at least one pebble to the target. A pebbling move of \(p\) pebbles across an edge with weight \(w\) leaves \(\lfloor pw \rfloor\) pebbles on the next vertex. We find the weighted pebbling numbers of stars, graphs with at least \(2|V|-1\) edges, and trees with given targets. We give an explicit formula for the minimum total weight required on the edges of a length-2 path, solvable with \(p\) pebbles, and exhibit a graph that requires an edge with weight \(1/3\) in order to achieve its weighted pebbling number.