The relevance of the concepts of fragility and of the glass transition temperature for the understanding of crystal nucleation and growth in glass-forming liquids is explored. It is shown that classical fragility can be relevant for the understanding of the crystallization behavior only if several severe conditions are fulfilled that are rarely met. By this reason, a new definition of liquid fragility is introduced shown to be able to reflect appropriately the maxima of crystal nucleation, growth, and overall crystallization rates. The origin of the previously reported correlations between glass transition temperatures and intensity of crystallization processes is specified. Further, interpreted in terms of classical nucleation theory, for a variety of oxide glassforming liquids, the thermodynamic barrier for homogeneous crystal nucleation exhibits at high undercooling an unusual increase with a decrease in temperature. Such behavior cannot be explained in terms of classical nucleation theory. Different directions of generalization of the classical theory are discussed, which may lead to a resolution of these problems. The realization of such generalization is considered to be a prerequisite for a generally deeper understanding of the complex phenomena occurring in melt crystallization with its rich field of applications.