Guanine-rich RNAs and DNAs from chromosomal telomeres and elsewhere that fold into guanine quadruplexes (G-quadruplexes), are found to complex tightly with porphyrins such as N-methylmesoporphyrin IX (NMM) and hemin [Fe(III) heme]. By themselves, these DNAs and RNAs are found to be efficient catalysts for porphyrin metallation. When complexed with hemin, under physiological conditions, these nucleic acids display robust peroxidase (one-electron oxidation), as well as peroxygenase (two-electron oxidation, or oxygen transfer) activity. These surprising catalytic properties, that frequently match the catalytic performance of natural peroxidase and P450 monooxygenase enzymes, have been the subject of significant mechanistic analysis, as well as having found utility in a wide range of biosensing and other applications. This review summarizes recent insights into a surprising yet fundamental property of many RNAs and DNAs, a property with undoubted ramifications for cellular oxidative disease, de novo hemoenzyme design, and our understanding of the evolution of early biocatalytic systems.