Covalent bonding is the natural consequence of the electron delocalization that occurs when two or more atoms form a stable molecule by the process of electron sharing. In an effort to present an easily accessible physical picture, covalent bonding is often described in terms of simple (but incorrect) electrostatic models that effectively ignore the quantum mechanical nature of electrons and the importance of delocalization. In this paper we reexamine the mechanism of covalent bonding, specifically with a view to its teaching, that starts with quantum theory and the interpretation of its predictions, such as electronic delocalization and the concomitant lowering of the electronic energy as bonding occurs. Indeed, delocalization is shown to be the central mechanism of covalent bond formation. These ideas are discussed in detail in the context of the simplest molecules: H2+ and H2. To help achieve our aims of teaching covalent bonding in a rigorous fashion from the start, we have developed a computer progra...