  Glassy solids exhibit a wide variety of generic thermomechanical properties,
ranging from universal anomalous specific heat at cryogenic temperatures to
nonlinear plastic yielding and failure under external driving forces, which
qualitatively differ from their crystalline counterparts. For a long time, it
has been believed that many of these properties are intimately related to
nonphononic, low-energy quasilocalized excitations (QLEs) in glasses. Indeed,
recent computer simulations have conclusively revealed that the
self-organization of glasses during vitrification upon cooling from a melt
leads to the emergence of such QLEs. In this Perspective Article, we review
developments over the past three decades towards understanding the emergence of
QLEs in structural glasses, and the degree of universality in their statistical
and structural properties. We discuss the challenges and difficulties that
hindered progress in achieving these goals, and review the frameworks put
forward to overcome them. We conclude with an outlook on future research
directions and open questions.
