The Bieri-Neumann-Strebel-Renz invariants of a group are a sequence of geometric objects that encode a great deal of information about certain subgroups of the group, including "finiteness properties" like finite generation and finite presentability. In general they are quite difficult to compute, and a full computation has been done only for very few "interesting" families of groups. I will discuss some of my results on the BNSR-invariants of the pure braid groups, and the implications for finiteness properties of their subgroups. In particular I will discuss some natural subgroups that are finitely generated but not finitely presented, finitely presented but not of type F_3, type F_3 but not F_4, and so forth.