The concept of the oxidation state of an atom in a chemical compound is formulated in terms of the quantum theory of many-electron systems and illustrated by calculations of oxidation numbers of calcium(I, II), phosphorus( 0-V), sulfur(0-VI), fluorine(-I), oxygen(-II), and krypton(0) in CaCB11H6Cl6, CaCB11H12, CaC5H5, Ca(C5H5)2, Ca@C60, P@C60, PF, PF3, PF5, POF3, P4O6, P4O10, SO, OSSO, SO2, SF2, SF4, SF6, SOF4, SO2F2, S3O9, and Kr@C60molecules. In [PO4]3-@aqua, [P3O9]3-@aqua, [PF6]-@aqua, [SO3]2-@aqua, and[SO4]2-@aqua ions, high oxidation states of phosphorus(V) and sulfur(IV, VI) are realized.