Population models in the birth-and-death style tradition have the unpleasant (and usually not advertised) implication that individuals do not age: It follows from the Markov properties of the whole population that life spans must be exponentially distributed and reproduction occur as splitting or in a Poisson process. This can be remedied only in parts (and at a high esthetical cost) by assuming more complicated Markovian properties in real time, like the age and parity dependent models of demography. Instead, if there is a sensible Markov structure in population growth, it resides in the pedigree, daughters inheriting genotypes from their mothers and being independent of their ancestors, given these types. This idea is used to define general branching processes and to analyze their properties: extinction, growth and asymptotic composition. The results are used to interpret the hypothesis of a molecular clock of mutations in biological evolution.