The book is an introduction to the problems and methods of numbering theory - an evolving section of algorithm theory. As far as the author knows, for the first time the idea of ​ ​ a systematic study of numbered sets was expressed by A. N. Kolmogorov in the mid-fifties. The implementation of this idea for computable numbering at that time was taken up by V. A. Uspensky. In parallel, a number of foreign mathematicians (Rayet, Dekker, Myhill, Friedberg, Lachlan, La Combe, Pur-El, etc.) also studied various issues related to computable numbering.