Braided vector spaces are vector spaces with an automorphism of their tensor square satisfying a certain equation. They produce natural representations of the braid group. Many problems in number theory and arithmetic statistics can be answered in the function field setting if one has sufficient information about the cohomology of certain covers of configuration space. These cohomology groups can often be computed as the group cohomology of representations of the braid group arising from braided vector spaces. The course introduces these topics and explains the connections between them.