Molecular conductance junctions, in which single molecules or small groups of molecules act as conductors of electrical current between two electrodes, have become an area of intense activity, driven by the intensive ongoing research and development effort in nanoscience and nanotechnology and by the growing expectations that molecules can become basic components of future electronic devices. At the same time this new point of view raises new fundamental questions, some of which are the subject of my research. This talk will provide an overview of our present understanding of the problem of molecular conductance. I will discuss different approaches and mechanisms, the relationship between conductance and the fundamental chemical kinetics of electron transfer, the relationship between junction structure and its conduction properties, the significance of thermal interactions, and some of the challenging areas not yet fully addressed.