The theory of Grobner Bases is a key computational tool for studying polynomial ideals. This theory has been introduced and developed by Buchberger in 1965. Whereas the original notion of Grobner Bases has been introduced over fields, we can extend it to the noetherian rings, especially PIDs and Galois rings. In 1978, Trinks has extended this notion from the field case to the ring case. Some mathematicians like Buchberger, Kandri-Rodi, and Kapur have established a specific notion over integer ring. By these achievements, we can define two kinds of Grobner Bases over rings: weak and strong Grobner Bases. We will first discuss weak Grobner Bases. Then, we will discuss strong Grobner Bases over PIDs and Euclidean Domains. In order to compute a weak Grobner Bases, we need a division algorithm and the generalized S-polynomial. As it mentioned, the notion of strong Grobner Bases will be defined over some special rings. In 2003, Daniel Lichtblau has introduced two extended Buchberger’s criteria to reduce the number of reductions to zero in computing strong Grobner Bases over Euclidean Domains. In the sequel, we will discuss these two criteria. Following [3], we have successfully adapted the update algorithm to implement these criteria. The results of this implementation have been shown in this thesis. The interesting -tiling problem is presented in this thesis as an application of strong Gr bner bases over integers. Finally, the last chapter is devoted to the computation of the Grobner Bases over Glois rings. This chapter will be finished by the decoding problem of alternant codes over Galois rings as an application of Grobner bases.