The properties of levels of charmonium--the bound system consisting of the charmed quark $\bar c$ and antiquark $\bar c$ -- are considered. A brief review is given of the experimental data on the different levels of charmonium, and the classification of the states and their decays are discussed. Of the latter, radiative transitions between levels and the annihilation of levels of charmonium to give photons (or lepton pairs), and also light hadrons ($\pi$, $\eta$ and $K$ mesons), are paid the most attention. Such decays have fundamental significance, inasmuch as they are connected in the most direct manner with the properties of quarks and their interactions. The theoretical foundation of the review is quantum chromodynamics--the theory of the interaction of colored quarks and gluons. The review contains the results of calculations performed in the framework of quantum chromodynamics and pertaining to the annihilation decays of charmonium levels and also to other phenomena: photoproduction of charmed particles, leptonic decays of charmed particles, and nonleptonic decays of strange particles.