Representations of Galois groups

We can learn a lot about an integral domain by studying the Galois group of its fraction field. These groups are generally quite complicated and hard to understand, but their representations, so-called Galois representations, contain more easily accessible information. These also play the lead in many important theorems and conjectures of modern maths, such as the Modularity theorem and the Langlands programme. In this talk we give a quick introduction to Galois representations, motivated by lots of examples aimed at a general algebraist audience, and talk about some open problems.