A collection of linked circles in $R^3$ is called a Brunnian link if the circles become unlinked when any one of them is removed. We describe how to formulate this phenomenon as a statement about groups and homomorphisms, interpreting the circles as the paths of a set of non-colliding particles in the plane, or equivalently in terms of configuration spaces and their fundamental groups. From there we generalize the idea to fundamental groups of complements of complex hyperplane arrangements. We report on recent work of the author with Daniel Cohen and Richard Randell, describing this subgroup of ``Brunnian elements" in such arrangement groups, and illustrate how it can be used to make qualitative statements relating to long-standing fundamental questions about these groups, e.g., the existence of elements of finite order.