A class of problems in quantum information theory, having simple formulation but still resisting general solution, concerns additivity properties of various quantities characterizing quantum channels, notably the "classical capacity", the "minimal output entropy" and the "entanglement of formation". All known results, including numerical work, are consistent with the conjecture that these quantities are additive with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity cannot be increased by using entangled inputs of the quantum channel. In this talk we review the current status of the additivity/multiplicativity problems, discuss relations between them and give a survey of relevant results and approaches.