Born in St. Petersburg, the son of the sculptor R. I. Bach. Initially studied with his father, then at the school of St. Anna in St. Petersburg. Was an auditor in the Imperial Academy of arts (1879-1885) at the sculpture Department at A. R. von Bock, N. A. Laveretskogo, I. Podosenova. Attended classes at the house of p. P. Chistyakov (1884). For the relief of the "Elf" received a small incentive medal (1882), portrait of P. P. Chistyakov – a great incentive medal and the title of class artist of the 3rd degree (1885). For the series of sculpted portraits and reliefs, presented at the exhibition of the Imperial Academy of arts (1886), received the title of class artist of 2 nd degree (diploma of the Imperial Academy of arts has not received, as not passed the whole course of science). He taught in the sculpture class at the Academy of arts in RSOPH (1886-1890) and at the architectural Department, class modeling vhu at IAH (1896-1917). The statue of the "Genius of art" and "for fame in the artistic field" was awarded the title of academician (1891). Professor (1900), member of the Imperial Academy of arts (1906). The most significant works are: the monument "Pushkin-Lyceum" in Tsarskoe Selo (1899), the monument to M. I. Glinka in St. Petersburg (1903-1906) and the monuments to Peter I in Tallinn (1909) and in Tula (1910). In the portrait genre R. R. Bach created a gallery of sculptures of figures of Russian culture. After the death of his elder brother (N. R. Bach), he spent 15 years were sent to KCLS their models for replication.