This thesis deals with the deepest descent methods applied on solving polynomials. The deepest descent methods are iterative methods in which new iteration point is the global minimizer of the chosen norm along the search direction. The methods differ with respect to the choice of the search direction and norm. Although the common choice of the norm is the Euclidean norm, the use of the max-norm is analysed in the thesis as well. The search direction can be any standard direction such as Newton direction, the steepest descent direction, the coordinate axes, or a combination of standard search directions. Furthermore, two new search directions are analysed: the Gauss-Seidel direction and the LP direction, which an A numerical experiment was performed to compare the methods’ performance. First, a collection of more than a100 polynomial systems from the relevant sources was compiled. Then, the success rate for each method and problem was calculated, whereby success rate was defined as the percentage of the convergent iteration processes which start with the initial point from any set S. The three indices, based on the success rate, were used for the comparison of the methods. The deepest descent methods were compared with two benchmark methods, the MinPack and the NLEQ, and the results suggest that nearly all the deepest descent methods had a better success rate than the benchmark methods. The research conducted in this thesis has shown that the deepest descent methods can be successfully used for fast and efficient solving of the polynomial systems which is consequently its main contribution to science.