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This article consists of two parts. The first part presents a detailed history of the long-term joint work (1960)(1961)(1962)(1963)(1964)(1965)(1966)(1967)(1968) of the author and P.S. Novikov on the proof of the infiniteness of the free Burnside groups B(m, n) for odd periods n ≥ 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 ≥ 101 the lower bound on the odd periods n for which one can prove the infiniteness of the free periodic groups B(m, n). We plan to publish a full proof of this new result in the journal Russian Mathematical Surveys.The Burnside problem on periodic groups was posed in 1902 by the British mathematician W. Burnside in the following form (see [27]):Let a 1 , a 2 , . . . , a m be independent elements generating a group G such that the relation x n = 1 holds for any element x ∈ G, where n is a given integer. Is the group G thus defined always finite? And if so, then what is the order of G?The maximal group presented by m generators and the identity relation x n = 1 is called the free Burnside group of rank m and period n. It is denoted by B(m, n) = a 1 , . . . , a m | x n = 1 .During several decades many eminent algebraists were working on this problem. The positive answer to Burnside's main question is known only for very small values of the exponent n. W. Burnside [27] proved the finiteness of the groups B(m, n) for any m and n ≤ 3; I.N. Sanov in [56] proved the same for the period n = 4; and M. Hall in [34] proved it for n = 6.Finiteness was also proved for all finitely presented linear periodic groups:• First Burnside himself in [28] proved finiteness for linear groups with uniformly bounded periods of all elements. • Then I. Schur [59] proved that the same is true for all finitely presented linear periodic groups.We should mention that among the most important open questions still are the problems of finiteness of the groups B(2, 5) and B(2, 8).A negative solution to the Burnside problem was first published in 1968 in the series of joint papers by P.S. Novikov and the author [49]. The main result of these joint papers was the following Novikov-Adian theorem. The free periodic groups B(m, n) of odd exponents n ≥ 4381 are infinite for all m > 1.