The phenomenology of the glass transition and the associated behavior in the near liquid and glassy states are detailed, including the cooling rate dependence of the glass transition, Kovacs' three signatures of structural recovery, and enthalpy overshoots. Dynamics in the liquid regime just above Tg and the associated temperature dependences are also covered since this behavior is important to understanding the glassy dynamics. The current models of structural recovery and their shortcomings are presented. A number of important unanswered questions are discussed, including how the relaxation time in the glassy state depends on structure, the relationship between the evolution of different properties, the resolution of the Kauzmann paradox, and the behavior of the equilibrium relaxation time below Tg. New experimental approaches are needed to make breakthroughs, such as two that are described: one involving 20 Ma amber to test whether the Vogel temperature dependence continues for the equilibrium state below Tg and another involving an ideal polymer/pentamer mixture to obtain the entropy of the liquid far below TK in a test of the Kauzmann paradox. An unexplored regime of glassy behavior, characterized by ultrastability, high density, and low fictive temperature, is identified, and experiments to understand the material behavior in this region are motivated.