Abstract: Ever since their first description in the 1965 PhD thesis of Bruno Buchberger, Gr$\mathrm{\ddot{o}}$bner bases have been an important tool for computational algebra. We can view Gr$\mathrm{\ddot{o}}$bner bases as a nonlinear version of Gaussian Elimination or a multivariate version of Euclid's Algorithm. They allow to answer problems in ideal theory, polynomial system solving, algebraic geometry, homological algebra, graph theory, diophantine equations and many other areas.

In this talk we will discuss the mathematical definition of Gr$\mathrm{\ddot{o}}$bner bases of polynomial ideals, computation of Gr$\mathrm{\ddot{o}}$bner bases with Buchberger's algorithm, conversion of Gr$\mathrm{\ddot{o}}$bner bases using the FGLM algorithm and the Gr$\mathrm{\ddot{o}}$bner walk, generalisations of Gr$\mathrm{\ddot{o}}$bner bases beyond commutative polynomials, and a selected number of applications including ideal comparison as well as symbolic summation.