In 2000, answering a question by R. Pandharipande, A. Okoun'kov proved that a generating function for connected double Hurwitz numbers
satisfies Toda equations. Since then, a number of other solutions to integrable hierarchies, whose coefficients are answers to various
enumerative problems, has been constructed in the work of I. Goulden, D. Jackson, J. Harnad, P. Zograf, and others. The talk will
present a review of modern approaches to constructing such formal solutions. The relationship between these approaches and
combinatorics of symmetric groups and their representations will be explained. Applications of the results to constructing
efficient computations in enumeration of various kinds of maps will be given.