The evolution (sensu lato) of the cosmos can be divided in three phases: cosmological evolution (sensu stricto), biological evolution and cultural evolution. Analogies between biological and cultural evolution date from the nineteenth century although it is only in the past two decades that so-called cultural evolution research has exploded. By contrast, comparisons between cosmological evolution and either biological or cultural evolution are uncommon. Here, we compare these three kinds of evolution and try to delineate their common grammar. Do their structure and underlying dynamics have characteristics in common? We believe that this is indeed the case and also that this Universal Grammar of Evolution can and should be used as a heuristic template in the study of these three kinds of evolution. Furthermore, this provisional template also might help scholars in their attempts to sketch the future course of a possible fourth kind of evolution, that of artificial intelligence and intelligent machines.