This master thesis has been prepared based on the \\cite{A}, \\cite{5A} and \\cite{Gao} . Let $P= k[x_{1}, \\ldots ,x_{n} ]$ be the polynomial ring in $ n $ variables over a field $ k$ . The vanishing ideal with respect to a set of points $ \\mathbb{X}= \\{p_{1}, \\ldots ,p_{m} \\} \\subseteq k^{n}$ is defined as the set of all elements in $P$ that are zero on all of the $ p_{i}$ 's (this set is an ideal of P and is denoted by $ I( \\mathbb{X} )$ ). The problem that we address in this thesis to compute the reduced Gr{\\ quot;o}bner basis for the vanishing ideal of any finite set of points, under any given monomial order. A polynomial time algorithm for this problem was first given by Buchberger and M{\\ quot;o}ller (1982) \\cite{53} , and significantly improved by Marinari, M{\\ quot;o}ller and Mora (1993) \\cite{510} , and Abbott, Bigatti, Kreuzer and Robbiano (2000) \\cite{51} . These algorithms perform Gauss elimination on a generalized Vandermonde matrix and have a polynomial time complexity.