The phenomenon of bacterial persistence, also known as non-inherited antibiotic tolerance in a part of bacterial populations, was described more than 70 years ago. This type of tolerance contributes to the chronization of infectious diseases, including tuberculosis. Currently, the emergence of persistent cells in bacterial populations is associated with the functioning of some stress-induced molecular triggers, including toxin-antitoxin systems. In the presented review, genetic and metabolic peculiarities of persistent cells are considered and the mechanisms of their occurrence are discussed. The hypothesis of the origin of persister cells based on bistability, arising due to the non-linear properties of a coupled transcription-translation system, was proposed. Within this hypothesis, the phenomenon of the bacterial persistence of modern cells is considered as a result of the genetic fixation of the phenotypic multiplicity that emerged in primitive cells in the process of neutrally coupled co-evolution (genetic drift of multiple neutrally coupled mutations). Our hypothesis explains the properties of persister cells, as well as their origin and "ineradicable" nature.