The demand has increased for sophisticated molecular tools with improved detection limits. Such molecules should be simple in structure, yet stable enough for clinical applications. Nucleic acid aptamers (NAAs) represent a class of molecules able to meet this demand. In particular, aptamers, a class of small nucleic acid ligands that are composed of single-stranded modified/unmodified RNA/DNA molecules, can be evolved from a complex library using Systematic Evolution of Ligands by EXponential enrichment (SELEX) against almost any molecule. Since its introduction in 1990, in stages, SELEX technology has itself undergone several modifications, improving selection and broadening the repertoire of targets. This review summarizes these milestones that have pushed the field forward, allowing researchers to generate aptamers that can potentially be applied as therapeutic and diagnostic agents.