Micromechanics-based homogenization has been employed extensively to predict the effective properties of technologically important composites. In this review article, we address its application to various physical phenomena, including elasticity, thermal and electrical conduction, electric, and magnetic polarization, as well as multi-physics phenomena governed by coupled equations such as piezoelectricity and thermoelectricity. Especially, for this special issue, we introduce several research works published recently from our research group that consider the anisotropy of the matrix and interfacial imperfections in obtaining various effective physical properties. We begin with a brief review of the concept of the Eshelby tensor with regard to the elasticity and mean-field homogenization of the effective stiffness tensor of a composite with a perfect interface between the matrix and inclusions.We then discuss the extension of the theory in two aspects. First, we discuss the mathematical analogy among steady-state equations describing the aforementioned physical phenomena and explain how the Eshelby tensor can be used to obtain various effective properties. Afterwards, we describe how the anisotropy of the matrix and interfacial imperfections, which exist in actual composites, can be accounted for. In the last section, we provide a summary and outlook considering future challenges.