Buchberger’s algorithm is the standard algorithm used to compute Grobner bases, which are foundational objects in computational algebra. There are many choices and strategies involved in implementations of Buchberger’s algorithm, and these choices can make dramatic differences in both the time and memory required for the computation. In this talk we will define Grobner bases, introduce Buchberger’s algorithm, and consider the effects of various choices throughout the algorithm, particularly in the strategy for pair selection.