The emergence of catalysis in early genetic polymers such as RNA is considered a key transition in the origin of life, pre-dating the appearance of protein enzymes. DNA also demonstrates the capacity to fold into three-dimensional structures and form catalysts in vitro. However, to what degree these natural biopolymers comprise functionally privileged chemical scaffolds for folding or the evolution of catalysis is not known. The ability of synthetic genetic polymers (XNAs) with alternative backbone chemistries not found in nature to fold into defined structures and bind ligands raises the possibility that these too might be capable of forming catalysts (XNAzymes). Here we report the discovery of such XNAzymes, elaborated in four different chemistries (arabino nucleic acids, ANA; 2'-fluoroarabino nucleic acids, FANA; hexitol nucleic acids, HNA; and cyclohexene nucleic acids, CeNA) directly from random XNA oligomer pools, exhibiting in trans RNA endonuclease and ligase activities. We also describe an XNA-XNA ligase metalloenzyme in the FANA framework, establishing catalysis in an entirely synthetic system and enabling the synthesis of FANA oligomers and an active RNA endonuclease FANAzyme from its constituent parts. These results extend catalysis beyond biopolymers and establish technologies for the discovery of catalysts in a wide range of polymer scaffolds not found in nature. Evolution of catalysis independent of any natural polymer has implications for the definition of chemical boundary conditions for the emergence of life on Earth and elsewhere in the Universe.