The purpose of this paper is to review two mathematical models: one for the formation of homochiral polymers from an originally chirally symmetric system; and the other, to show how, in an RNA-world scenario, RNA can simultaneously act both as information storage and a catalyst for its own production. We note the similarities and differences in chemical mechanisms present in the systems. We review these two systems, analysing steady states, interesting kinetics and the stability of symmetric solutions. In both systems we show that there are ranges of parameter values where some chains increase their own concentrations faster than others.