A system with artificial intelligence usually relies on symbol manipulation, at least partly and implicitly. However, the interpretation of the symbols - what they represent and what they are about - is ultimately left to humans, as designers and users of the system. How symbols can acquire meaning for the system itself, independent of external interpretation, is an unsolved problem. Some grounding of symbols can be obtained by embodiment, that is, by causally connecting symbols (or sub-symbolic variables) to the physical environment, such as in a robot with sensors and effectors. However, a causal connection as such does not produce representation and aboutness of the kind that symbols have for humans. Here I present a theory that explains how humans and other living organisms have acquired the capability to have symbols and sub-symbolic variables that represent, refer to, and are about something else. The theory shows how reference can be to physical objects, but also to abstract objects, and even how it can be misguided (errors in reference) or be about non-existing objects. I subsequently abstract the primary components of the theory from their biological context, and discuss how and under what conditions the theory could be implemented in artificial agents. A major component of the theory is the strong nonlinearity associated with (potentially unlimited) self-reproduction. The latter is likely not acceptable in artificial systems. It remains unclear if goals other than those inherently serving self-reproduction can have aboutness and if such goals could be stabilized.