Complexity theory, proofs and approximation

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