Giuseppe Peano (1858-1932) and his collaborators (C. Burali-Forti, A. Padoa, M. Pieri, G. Vailati et al.) working at the end of the nineteenth and the beginning of this century, played an important role in the development of a modern approach to mathematical logic, the foundations and the didactics of mathematics. The authors survey the main results obtained by Peano and his school and discuss many interesting questions, e.g., why Peano's school did not join new and more modern fields of the foundational research. Numerous historical facts concerned with Peano's school contributions to the development of mathematical logic and the axiomatic method, as well as the foundations of geometry and arithmetic are analysed in the article.