Some years after the publication of B. Buchberger's fundamental paper on Grobner bases for ideals in commutative polynomial rings over fields W. Trinks published a natural generalization to polynomial rings over commutative noetherian rings . He translated in a natural way the notions of S-polynomial and of reduction from the field case to the ring case. Another access to Grobner bases over rings was proposed in Buchberger , Kandri-Rody and Kapur , Moller . There are many reasons to study Grobner bases in polynomial ring over a ring, here is one of them: In order to speed up the computation of Grobner bases over the field of rational numbers, one can try to use residue- stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" o:preferrelative="t" o:spt="75" coordsize="21600,21600" ,, and . Grobner bases in (non-commutative) rings of differential operators (e.g. the Wely-algebra) were introduced in Galligo and Castro . Improvements of the algorithm to compute these Grobner bases have recently been made in Winkler and Zhou . This thesis presents a unified approach to Grobner bases of left-ideals in a Many investigations have been done on Grobner basis in rings of differential operators, but the coefficients are in fields (of rational functions), rings of power series, or rings of polynomials over a field. As in Insa and Pauer's paper , the rings of coeffiecient in this paper are general commutative rings, which is the main difference from other existing works. In Insa and Pauer's paper, the results of Buchberger on Grobner basis in polynomial rings have been extended to the theory of Grobner basis for differential operetors. Further, we study a criterion which was presented in to determine if a set of differential operators is a Grobner basis, and also, basic method for computing the Grobner basis. Pauer generalized the theory to a For computing the Grobner basis of a set of differential operators, insted of computing the generators of the syzygy module generated by their initials (like the conventional methods to compute Grobner bases), Insa and Pauer's method needs to compute the generators of many syzygy modules generated by their leading coefficients (which are polynomials). Thus, Insa and Pauer's method leads to many unnecessary computations. In order to improve the efficiency, Zhou and Winkler proposed some techniques to reduce the computations on the syzygies . In this thesis we study also this technic and implement it in Maple to investigate its efficiency. In this thesis, based on works in , a new criterion is proposed for computing Grobner basis in the ring of differential operators with coefficients in a general commutative ring. In new method, it suffices to consider the generators of the syzygy module in a commutative ring which is deduced from the ring of differential operators. With these generators, a new criterion is proposed to determine if a set of differential operators is a Grobner basis. This new result generalizes original Insa-Pauer theorem such that their theorem can be extended naturally to the rings that preserve the same fact. Then the proposed criterion also leads to an efficient method for computing Grobner bases in the rings of differential operators. This new method computes fewer S-polynomials than those in Insa and Pauer's method as well as Zhou and Winkler's improved version. So it is not surprising that this new method will have better efficiency than others.