The principle of Darwinian evolution has been applied in the laboratory to nucleic acid molecules since 1990, and led to the emergence of in vitro evolution technique. The methodology of in vitro evolution surveys a large number of different molecules simultaneously for a pre-defined chemical property, and enrich for molecules with the particular property. DNA and RNA sequences with versatile functions have been identified by in vitro selection experiments, but many basic questions remain to be answered about how these molecules achieve their functions. This dissertation first focuses on addressing a fundamental question regarding the molecular recognition properties of in vitro selected DNA sequences, namely whether negatively charged DNA sequences can be evolved to bind alkaline proteins with high specificity. We showed that DNA binders could be made, through carefully designed stringent in vitro selection, to discriminate different alkaline proteins. The focus of this dissertation is then shifted to in vitro evolution of an artificial genetic polymer called threose nucleic acid (TNA). TNA has been considered a potential RNA progenitor during early evolution of life on Earth. However, further experimental evidence to support TNA as a primordial genetic material is lacking. In this dissertation we demonstrated the capacity of TNA to form stable tertiary structure with specific ligand binding property, which suggests a possible role of TNA as a pre-RNA genetic polymer. Additionally, we discussed the challenges in in vitro evolution for TNA enzymes and developed the necessary methodology for future TNA enzyme evolution.