While the DNA sequence is largely responsible for transmitting phenotypic traits over evolutionary time, organisms are also considerably affected by phenotypic variations that persist for more than one generation, with no direct change in the organisms' DNA sequence. In contrast to genetic variation, which is passed on over many generations, the phenotypic variation generated by nongenetic mechanisms is difficult to study due to the inherently limited life time of states that are not encoded in the DNA sequence, but makes it possible for the 'memory' of past environments to influence future organisms. One striking example of phenotypic variation is the phenomenon of bacterial persistence, whereby genetically identical bacterial populations respond heterogeneously to antibiotic treatment. Our aim is to review several experimental and theoretical approaches to the study of persistence. We define persistence as a characteristic of a heterogeneous bacterial population that is taken as a generic example through which we illustrate the approach and study the dynamics of population variability. The clinical and evolutionary implications of persistence are discussed in light of the mathematical description. This approach should be of relevance to the study of other phenomena in which nongenetic variability is involved, such as cellular differentiation or the response of cancer cells to treatment.