The behavior of glass-forming materials is examined with emphasis on the below-glass transition behavior. A major question that is related to the super-Arrhenius behavior of the dynamics of glass-forming systems is whether the apparent divergence at finite temperature continues below the kinetic or laboratory glass transition that is related to the limits of measurement and is standardized so that the material relaxation time is near 100 s. The problem arises because as the temperature decreases, the time scales required to reach equilibrium (or metastable equilibrium) become geologically long. Yet the apparent finite temperature divergence is fundamental to many theories of glasses; therefore, it becomes essential to find ways to finesse the extreme time scales related to the so-called Kauzmann paradox to bring new information to the ongoing conversation concerning the existence or not of an ideal glass transition at either the Kauzmann temperature or the Vogel-Fulcher-Tammann temperature. After describing the framework of the glassy state that is formed by the early ideas of a fictive temperature, we examine the use of extremely low fictive temperature glasses as a means to potentially get around the long time-scale problem. The challenge is to find ways to create such glasses and measure their properties. In addition to looking at the dynamic behavior of a 20-million-year-old amber and a vapor-deposited amorphous perfluoropolymer whose fictive temperature was the same as the Kauzmann temperature for the material, we also examine the possibility of directly testing the thermodynamics of an ideal glass transition by making athermal solutions of a poly(a-methyl styrene) and its pentamer, where we find that the entropy surface determined from extrapolation of the heat capacity to zero pentamer shows no distinct transition at as much as 180 K below the Kauzmann temperature. The significance of the dynamics of the stable glasses and the thermodynamics of the polymer solutions is discussed in terms that challenge the idea of an ideal glass transition. We also look in more detail at the ability to use vapor deposition to make ethylbenzene, a small-molecule organic, into an ultra-stable glass with a fictive temperature that is possibly below the Kauzmann temperature of this material. We end with remarks on the question of decoupling of different relaxation mechanisms as something not treated by current theories of glass, and we consider some open questions related to the fact that the glass transition remains an unresolved and important problem.