We re-investigate the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices, by re-examining their correlations to operations with univariate polynomials. Then we show some natural extensions of such classes of matrices based on the correlations to multivariate polynomials. We describe these correlations in terms of the associated operators of multiplication in the polynomial ring and its dual, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues and relations between them are studied. Finally, we show some applications of structured matrices to root finding problems for a system of multivariate polynomial equations, where these matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these major problems.