The structural glass transition is often regarded as purely a problem of the classical theory of liquids. The dynamics of electrons enters only implicitly, through the interactions between ionic cores or molecules. Likewise, zero-point effects tied to the atomic masses hardly affect the typical barriers for liquid rearrangements. Yet, glasses do exhibit many quantum phenomenaelectronic, optical, and cryogenic peculiarities that seem to have universal characteristics. These anomalies of the glassy state are uncommon or strongly system dependent in crystals and amorphous solids not made by a quasi-equilibrium quench of a melt. These clearly quantum phenomena include midgap electronic states in amorphous semiconductors, the two-level systems, and the Boson peak. Here, we discuss how these quantum phenomena found in glasses are not merely consequences of any kind of disorder but have universal characteristics stemming from the structural dynamics inherent in the glass transition itself. The quantum dynamics at cryogenic temperatures and electronic dynamics are related to the transition states for relaxational motions above the glass transition temperature, which are partially frozen when the sample is quenched.