This thesis is an extension (and generalization) of the works done by Agnes Szanto [30,31] on solving polynomial systems using subresultants . Solving systems of polynomial equations is one of the fundamental challenges of computational algebraic geometry. On the other hand, it has many applications in sciences and engineering. In this thesis, we use some methods for solving zero dimensional polynomial equations (systems having a finite number of common roots). Further, multivariate resultant is a fundamental tool for solving polynomial systems in computer algebra. The resultant of a polynomial system has many important properties for the geometry of the variety that the system defines. Indeed, the resultant is an algebraic condition in terms of the coefficients of the given system of polynomials which is satisfied if and only if the system has a nontrivial common solution. A number of methods exist for constructing resultant matrices; i.e. matrices whose determinant is the resultant or, more generally, a nontrivial multiple of it. These matrices represent the most efficient way for computing the resultant polynomial and for solving systems of polynomial equations by means of the resultant method. An example of a matrix that gives precisely the resultant is the determinant of the coefficient matrix of linear polynomials, or the Sylvester matrix of a pair of polynomials. A generalization of this result can be used to decide whether a system of homogeneous equations in variables has a solution or not. This multipolynomial resultant can be used to eliminate variables from three or more equations and it is a surprisingly powerful tool for finding solutions of equations. Another method for solving systems of polynomial equations described in this thesis, is subresultant . The primary concern of the present paper is to find efficient methods to compute multivariate generalizations of the univariate subresultants and to use the resulting subresultant matrices to solve polynomial systems. For this, we describe a general framework for subresultant matrices and study the so called subresultant property fo r a given ideal. Furthermore, we describe the subresultant method to compute certain simple elements in the given ideal. We also introduce the notion of strong subresultant property with respect to a set of polynomial systems. We prove that if a matrix has the strong subresultant property with respect to some set of polynomial systems, then for any given system in this set, the polynomials computed via the subresultant method generate the same affine ideal. Next we demonstrate that the solution of the system of polynomials that was computed by the subresultant method is usually easier than solving the original system. Also, we show how to derive a triangular representation of their solutions or express the coordinates of the solutions as eigenvalues of multiplication matrices: all we need is to set up small matrices from the coefficients of the polynomials and then to take determinants. Our general framework allows to use matrices of significantly smaller size than previous methods which can improve the efficiency of the computations. The main result of this thesis is that simple modifications of matrices satisfy the strong subresultant properties with respect to set of homogeneous polynomial systems do not have roots at infinity.