Minimal self-replicating systems typically consist of three components: a product molecule, and two substrate molecules that become joined to form another product molecule. An important characteristic of self-replicating systems is the ability of the product to catalyze the formation of additional product, resulting in autocatalytic behavior. Recent advances in the area of self-replication have led to improved efficiency of autocatalysis, both by increasing the fraction of product molecules that can participate in further rounds of replication, and by improving the efficiency of the catalysts themselves. This review analyzes chemical self-replicating systems that have been developed to date and discusses ongoing challenges in this area of research.