In the first part of this thesis, we recall the notions and definitions related to the theory of Gr?bner bases. We present also some application of Gr?bner bases in the context of the thesis. Then we show that the Gr?bner basis is not the complete concept in the polynomial ideal theory. We show that Gr?bner basis is not the wholly sacred generating system of an ideal. The following two problems highlight a couple of serious obstacles in this direction: First problem: A small change in the coefficients of polynomials in an ideal has led to a big change in the Gr?bner basis of the ideal and in the associated vector space basis, although the zeros of the system have not changed much. Second problem: Gr?bner bases break the symmetry! It is more natural to compute in the vector space associated to the quotient ring of an ideal using a symmetric basis. Therefore we would like to find another system of generators of an ideal which has properties similar to a Gr?bner basis. we describe then the theory of border bases of zero-dimensional polynomial ideals. In the most cases, border bases behave numerically more stable than Gr?bner bases. The decision to use border bases to describe vanishing ideals of sets of empirical points is due to two main reasons: border bases have always been considered a numerically stable tool. Furthermore, it is easy to study the structure of border bases. Indeed, we can determine completely the support of the polynomials in the basis, once a suitable order ideal is chosen. Let X be a set of points whose coordinates are known with limited accuracy; There exist hypersurfaces passing through all points of X. The polynomials defining these hypersurfaces are called the separators of X. They can be used to solve the interpolation problem for X. Algorithmically, the main task is to compute the vanishing ideal of an affine point set from the coordinates of the points. In principle this task can be solved using the Gr?bner basis method. A much better approach is the Buchberger-M?ller algorithm which performs linear algebra. Moreover, the Buchberger-M?ller algorithm can be modified to produce the separators. We present also a method for computing “structurally stable” border bases of ideal of points whose coordinates are affected by errors. Our aim is to give a characterization of the vanishing ideal I ( X ) independent of the data uncertainty where X is a set of points. We present a method to compute, starting from X , a polynomial basis B of I ( X ) which exhibits structural stability, that is, if X is any set of points differing only slightly from X , there exists a polynomial set B structurally similar to B , which is a basis of the perturbed ideal I ( X ). The border bases can be generalized to general polynomial ideals (zero and positive dimensional ideals). A new algorithm to compute a border basis for a given polynomial system with respect to a specified term order is presented. This algorithm computes a border basis as well as a reduced Gr?bner basis for the polynomial system. A particular eigen vector method to find some components of positive dimensional polynomial system is sketched too.