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The complexity of the Black-White Pebbling Game has remained open for 30 years. It was devised to capture the power of non-deterministic space bounded computation. Since then it has been applied to problems in diverse areas of computer science including VLSI design and more recently propositional proof complexity. In this paper we show that the Black-While Pebbling Game is PSPACE-complete. We then use similar ideas in a more complicated reduction to prove the PSPACE-completeness of Resolution space. The reduction also yields a surprising exponential time/space speedup for Resolution in which an increase of 3 units of space results in an exponential decrease in proof-size.