Chromonic liquid crystals form when molecules aggregate into anisotropic shapes at high enough density to promote orientational order. There is strong evidence that in some systems the aggregates are simple columnar stacks of molecules and that the aggregation process is governed by free energy changes that are independent of the size of the aggregate. Theoretically such systems should not possess a critical concentration or critical temperature for aggregation, and this is confirmed by experimental results. However, there is also excellent evidence that in one system a critical concentration or critical temperature for aggregation occurs. There is also good evidence that the aggregate structure can differ from a simple stack of molecules in several ways. For example, the evidence is strong that some aggregate structures consist of two molecules in a cross-section, while other structures are much larger and probably contain a good deal of water. The current state of knowledge of molecular aggregation and chromonic liquid crystals will be described from both a theoretical and experimental perspective. Characteristics that are common to all chromonic systems will be outlined, along with a careful discussion of the differences between such systems. The results of recent work will be described, along with an analysis of some of the most important unanswered questions.