ABSTRACT: The Busemann-Petty problem (posed in 1956) asks the following question. Suppose that K and L are two origin-symmetric convex bodies in n-dimensional Euclidean space such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L. Does it follow that the (n-dimensional) volume of K is smaller than the volume of L? The answer is negative in dimensions 5 and larger (it was proved during the period from 1975 to 1993 by individual counterexamples lowering the dimension from 12 to 5). The answer is affirmative for dim=3 (R.J.Gardner, 1994). Until very recently, the answer in the case dim=4 was considered to be negative, but several months ago G. Zhang revised the solution and showed that the opposite is true. We present a new approach to the Busemann-Petty problem, based on the use of the Fourier transform, which initiated the revision of the case dim=4, provided a large class of bodies K for which the answer is positive for every choice of L with larger sections, and put the production of counterexamples in dim=5 and higher on the industrial level. This approach also leads to a new simple solution to the problem (in all dimensions), and explains why the transition occurs between dim=4 and dim=5.