From 1994 to 2004, research in DNA computing was in its nascent stage, encompassing various aspects such as computational models, encoding, experimentation, and detection. Notably, during this period, the computational models were primarily enumerative in nature. These pioneering research outcomes not only laid the foundation for the deeper exploration of DNA computing but also provided a solid groundwork for RNA computing and, more broadly, the entire field of biological computing. This chapter focuses on the enumerative DNA computing models, selecting a subset of representative achievements for detailed introduction and in-depth analysis.