Many important transitions in evolution are associated with novel ways of storing and transmitting information. The storage of information in DNA sequence, and its transmission through DNA replication, is a fundamental hereditary system in all extant organisms, but it is not the only way of storing and transmitting information, and has itself replaced, and evolved from, other systems. A system that transmits information can have limited heredity or indefinite heredity. With limited heredity, the number of different possible types is commensurate with, or below, that of the individuals. With indefinite heredity, the number of possible types greatly exceeds the number of individuals in any realistic system. Recent findings suggest that the emergence and subsequent evolution of very different hereditary systems, from autocatalytic chemical cycles to natural language, accompanied the major evolutionary transitions in the history of life.