Operator product expansion, heavy quarks, QCD duality and its violations

The quark (gluon) -- hadron duality constitutes a basis for the theoretical treatment of a wide range of inclusive processes -- from hadronic \tau decays and R_{e^+e^-}, to semileptonic and nonleptonic decay rates of heavy flavor hadrons. Theoretical analysis of these processes is carried out by using the operator product expansion (OPE) in the Euclidean domain, with subsequent analytic continuation to the Minkowski domain. We formulate the notion of the quark (gluon) -- hadron duality in quantitative terms, then classify various contributions leading to violations of duality. A prominent role in the violations of duality seems to belong to the so called exponential terms which, conceptually, may represent the (truncated) tail of the power series. A qualitative model, relying on an instanton background field, is developed allowing one to get an estimate of the exponential terms. We then discuss a number of applications, mostly from heavy quark physics.