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Abstract: Multiple zeta values (MZVs), also known as multiple harmonic series, have been actively investigated for about a decade. The first wave of interest in the early 1990's, triggered mainly by the appearance of MZVs as coefficients in Kontsevich's knot invariant, resulted in proofs of a number of striking identities. MZVs also occur in perturbative quantum field theory, and new applications continue to appear. We will discuss in detail the newly discovered "cyclic sum identities" and the appearance of MZVs in mirror symmetry.