The theory of Gr¨obner bases is a key computational tool for studying polynomial ideals. This theory was introduced and developed by Buchberger in 1965 (see his PhD thesis [1]). His two criteria (to detect the useless critical pairs) and the implementation methods (see [2]) made the Gr¨obner bases a powerful tool to solve many important problems in polynomial ideal theory. Lazard in 1983 in [9] by using linear algebra techniques could compute Gr¨obner bases. In 1988, Gebauer and M¨oller have installed Buchberger’s two criteria on Buchberger’s algorithm in an efficient way (see [6] or [3] page 230). In 1992, M¨oller, Mora and Traverso in [12] using syzygies, improved computation of Gr¨obner bases. Faug`ere in 1999 and 2002 has proposed F4 and F5 algorithms in [4, 5] where F5 is the fastest algorithm for computing Gr¨obner bases. The concept of comprehensive Gr¨obner bases can be considered as an extension of Gr¨obner bases of polynomials over fields to polynomials with parametric coefficients. This extension can play an important rule in con- structive algebraic geometry, robotics, electrical network, automatic theo- rem proving and so on (see [10, 11, 13, 14] for example). Comprehensive Gr¨obner bases was introduced and constructed by Weispfenning in [17]. He has proved that any parametric polynomial ideal has a comprehensive Gr¨obner basis and has described an algorithm for computing it. In 1997 Kalkbrenner in [7] has studied the stability of Gr¨obner bases under spe cialization. Then Montes in [13] has proposed a more efficient algorithm for discussing parametric Gr¨obner bases (DisPGB). The main purpose of this thesis is to introduce the concept of comprehen- sive Gr¨obner bases and to present Weispfenning’s algorithm for computing these bases. In the sequel, we present a more efficient algorithm for com- puting comprehensive Gr¨obner bases proposed by Montes. Montes in his DisPGB algorithm has not explicitly used Buchberger’s criteria. In this thesis, we improve DisPGB algorithm by a non trivial use of Buchberger’s two criteria. Also, we propose a new strategy for the selection of polyno- mials in this algorithm. We have implemented DisPGB algorithm and the modified version of DisPGB in Maple 12 and we compare the performance of these two algorithms via some examples (this implementation is available in https://amirhashemi.iut.ac.ir/software.html).