Various problems of pure and applied sciences such as physics, chemistry, biology, engineering, economics, management sciences, industrial research and optimization can be studied in the unified frame work of the system of nonlinear equations. In this study, we develop several new iterative methods for solving a system of nonlinear equations by using different techniques, including quadrature technique, new decomposition method and variational iteration technique. We prove the convergence of the new methods. We obtain the upper bounds of the error of new methods. We also discuss the efficiency index of the proposed methods and compare it with some other well-known methods. We provide several examples for the implementation and performance of the new methods. Comparison with some other existing methods is also given. We also discuss some open research problems.