There is much interest in the dynamics of glass-forming systems above the glass transition temperature. However, in many applications the behavior of the systems needs to be understood in the glass-transition Tg range and particularly in the non-equilibrium state. We present a set of empirical observations for the dynamics of material behaviors near to, but below the glass transition for polymer glass formers and outline a "minimal" set of requirements that might be expected of computer simulations in the non-equilibrium glass near to Tg. The survey includes the kinetics and nonlinearity of the structural (volume or enthalpy) recovery of the glass and its impact on the mechanical response (physical aging). These are presented with the thought that computer simulations may well be able to provide insights to the origins of an extremely complicated set of experimental observations. Additionally, we present some results of glass formation of simple liquids constrained to nanometer size pores with the expectation that such experiments may be readily simulated because of the small number of molecules in such pores.