We have carried out B3PW91 and MP2-FC computational studies of dimethyl sulfoxide, (CH(3))(2)SO, and dimethyl sulfone, (CH(3))(2)SO(2). The objective was to establish quantitatively the basis for their high polarities and boiling points, and their strong solvent powers for a variety of solutes. Natural bond order analyses show that the sulfur-oxygen linkages are not double bonds, as widely believed, but rather are coordinate covalent single S(+)-->O(-) bonds. The calculated electrostatic potentials on the molecular surfaces reveal several strongly positive and negative sites (the former including sigma-holes on the sulfurs) through which a variety of simultaneous intermolecular electrostatic interactions can occur. A series of examples is given. In terms of these features the striking properties of dimethyl sulfoxide and dimethyl sulfone, their large dipole moments and dielectric constants, their high boiling points and why they are such good solvents, can readily be understood.