I report on recent numerical simulations designed to study the inner structure of cold-dark-matter (CDM) haloes. This work confirms and extends earlier indications that the mass profiles of CDM haloes are roughly independent of mass and that the dark-matter density appears to diverge in a central "cusp". Our simulations also show that CDM halo-density profiles get increasingly shallow inwards down to the innermost radius (r(min)) resolved by the simulations, with no evidence for convergence to a well-defined asymptotic value (beta(0)) of the logarithmic slope. The dark mass contained within r(min) places strong constraints on beta(0); cusps as steep as beta(0) approximately -1.5 are clearly ruled out in our highest resolution simulations, although beta(0) less, similar -1 is consistent with the numerical data. The circular-velocity curves of the simulated CDM haloes are generally consistent with the rotation curves of low-surface-brightness (LSB) galaxies in the literature, but there are also some clearly discrepant cases. This disagreement has been interpreted as excluding the presence of cusps, but I argue that it may just reflect the difference between circular velocity and rotation speed likely to arise in gaseous discs embedded within realistic triaxial haloes. The perceived challenge to the CDM paradigm stemming from LSB rotation curves may thus plausibly result from the complex three-dimensional structure of CDM haloes, and might be resolved without appeal to a drastic revision of the paradigm.