The properties of levels of charmonium—the bound system consisting of the charmed quark and antiquark —are considered. A brief review is given of the experimental data on the different levels ofcharmonium, and the classification of the states and their decays are discussed. Of the latter, radiativetransitions between levels and the annihilation of levels of charmonium to give photons (or lepton pairs), and also light hadrons (π,η and K mesons), are paid the most attention. Such decays have fundamentalsignificance, inasmuch as they are connected in the most direct manner with the properties of quarks andtheir interactions. The theoretical foundation of the review is quantum chromodynamics—the theory of theinteraction of colored quarks and gluons. The review contains the results of calculations performed in theframework of quantum chromodynamics and pertaining to the annihilation decays of charmonium levelsand also to other phenomena: Photoproduction of charmed particles, leptonic decays of charmed particles, and nonleptonic decays of strange particles.