Gröbner bases have become a fundamental and multi-purpose tool in computational algebra. Originally introduced to answer questions about ideals of multivariate (commutative) polynomials, they were subsequently generalized to several noncommutative settings, including noncommutative polynomials in the free algebra. We present the main constructions and
results of the theory of Gröbner bases in this setting and give several applications. We will see a lot of similarities but also some crucial differences to
the commutative case. Additionally, we touch on some advanced topics in the field of noncommutative Gröbner bases, that, on the one hand, aim at
improving the efficiency of computations, and, on the other hand, expand the applicability of the theory. More precisely, we discuss linear algebra reductions (as known from the F4 algorithm), signature-based algorithms (such as F5 or GVW), and Gröbner bases in free algebras over coefficient rings.
