A fundamental quantum theory of conductivity of spatially inhomogeneous systems in weak electro-magnetic fields has been derived using a two-time Green function (TTGF)-based technique that generalizes the original method due to Zubarev and Tserkovnikov (ZT). Quantum current and charge density evolution equations are derived in a linear approximation with regard to the field potentials. Explicit expressions for the longitudinal and transverse conductivity, and dielectric and magnetic susceptibilities have been derived in terms of charge density - charge density and microcurrent - microcurrent TTGFs. The obtained theoretical description and formulae are applicable to any inhomogeneous system, such as artificial molecules, atomic and molecular clusters, thin films, interfacial systems, etc. In particular, the theory is designed to predict charge transport properties of small semiconductor quantum dots (QDs) and wells (QWs), and is a significant step toward realization of a concept of virtual (i.e., theory-based, computational) synthesis of electronic nanomaterials of prescribed electronic properties.