  In this mostly survey paper, we investigate the resonance varieties, the
lower central series ranks, and the Chen ranks, as well as the residual and
formality properties of several families of braid-like groups: the pure braid
groups $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid
groups $vP_n$, as well as their `upper' variants, $wP_n^+$ and $vP_n^+$. We
also discuss several natural homomorphisms between these groups, and various
ways to distinguish among the pure braid groups and their relatives.
