In this review we discuss systems of self-replicating molecules in the context of the origin of life and the synthesis of de novo life. One of the important aspects of life is the ability to reproduce and evolve continuously. In this review we consider some of the prerequisites for obtaining unbounded evolution of self-replicating molecules and describe some recent advances in this field. While evolution experiments involving self-replicating molecules have shown promising results, true open-ended evolution has not been realized so far. A full understanding of the requirements for open-ended evolution would provide a better understanding of how life could have emerged from molecular building blocks and what is needed to create a minimal for...