In this review we shortly summarize in an accessible way the physical and nanoscale material science aspects of the promising field of molecular quantum cellular automata (QCA). QCA is a revolutionary paradigm in quantum electronics with promising application in the conceptually new computing scheme which can be referred to as “computing with molecules”. This scheme of devices have vitally important advantages compared with conventional schemes of quantum computing in which the binary information is stored in the eigenvectors of a two-level quantum system and therefore their practical application is strongly limited by the requirement of coherence. Molecular QCA promise nanometer-scale units with ultra-high device densities, as well as room temperature operation with extremely small heat release. Molecular materials provide at the same time options to control the key properties of the active molecules by chemical means. Hereunder we summarize the background of the electronic and vibronic problems in QCA cells represented by the tetrameric mixed-valence (MV) complexes and organic molecules proposed for implementation as four-site molecular QCA. We discuss the basic model of the molecular cell that include the following ingredients: 1) intracell Coulomb repulsion energy of the electrons in different electronic distributions among the redox sites; 2) electron transfer between redox sites which changes electronic distributions; 3) interaction of the itinerant electrons with molecular vibrations which is referred to as the vibronic interactions; 4) intercell Coulomb interaction by mean of which the binary information is transmitted from cell to cell. On the basis of this model we will study the cell polarization, adiabatic picture of the switching cycle describing interaction between the “input” and “output” molecular cells, non-linear cell-cell response function whose shape describes the action of molecular logic gates. Along with developed and current problems, we discuss new challenging trends of research in this fascinating area. © 2020 Wiley-VCH Verlag GmbH