The authors consider the core structures of quantised vortex lines in Bose and Fermi superfluid systems. They emphasise the result that, within the core of a singly quantised vortex line in a Bose superfluid, there spontaneously appears a condensate of boson pairs in a relative d-wave state, instead of the normal fluid. Also, in the case of the singly quantised Abrikosov vortex line in a superconductor, instead of normal electrons there emerges a novel state of superconductivity, in which a condensate of bosons made of four fermions appears. In particular, they discuss these core structures as further examples for the flaring-out of the vortex singularity into higher dimensions, in analogy with the extended-space topology of the superfluid ³He-B vortex core; they also relate this generic behaviour to the Laughlin state within the quantum Hall effect.