Molecular solids are generally highly insulating. The creation of conducting molecular solids proved to be a major scientific challenge. As in the case of Si technology, the challenge started as impurity doping in band insulators and then developed into highly doped polymers, which are not crystalline. More conducting materials in crystalline forms have been realized in charge transfer (CT) complexes with two different kinds of molecules, where electrons are transferred between them in solids. In such CT complexes, not only conducting, but also even superconducting systems were achieved in 1980 and today more than 100 different superconductors are known. The most remarkable achievement in this direction was the realization of a truly metallic state in molecular solids based on a single kind of molecule. These are called single component molecular metals (SCMM) and consist of a rich variety of electronic properties. In these conducting molecular solids, CT and SCMM, many interesting electronic properties resulting from mutual Coulomb interactions and electron-phonon interactions have been explored so far, and these will be reviewed briefly in this article from a theoretical viewpoint. Challenges to come, based on these achievements, are also discussed at the end of this review.