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We give a short introduction to some questions in complexity theory and proceed to describe some recent developments. In particular, we discuss probabilistically checkable proofs and their applications in establishing inapproximability results. In a traditional proof the proof-checker reads the entire proof and decides deterministically whether the proof is correct. In a probabilistically checkable proof the proof-checker randomly verifies only a very small portion of the proof but still cannot be fooled into accepting a false claim except with small probability.