New estimates of odd exponents of infinite Burnside groups

This article consists of two parts. The first part presents a detailed history of the long-term joint work (1960–1968) of the author and P.S. Novikov on the proof of the infiniteness of the free Burnside groups $\mathbf {B}(m,n)$ for odd periods $n\ge 4381$ and $m>1$ generators (Sections 1 and 2). In Sections 3–10 we survey several significant results obtained by the author and his successors using the Novikov–Adian theory and its various modifications. In the second part (Sections 11–15) we outline a new modification of the Novikov–Adian theory. The new modification allows us to decrease to $n \ge 101$ the lower bound on the odd periods $n$ for which one can prove the infiniteness of the free periodic groups $\mathbf {B}(m,n)$. We plan to publish a full proof of this new result in the journal Russian Mathematical Surveys.