The core-cusp problem is the oldest unhealed wound in the dark matter paradigm. Cold collisionless dark matter, left to cluster under gravity in N-body simulations, generically builds halos with steep central cusps, density rising as 1/r toward the center in the canonical Navarro-Frenk-White profile (Flores and Primack 1994; de Blok 2010). The prediction is robust precisely because the physics is so simple: collisionless particles and gravity, nothing to soften the middle.
The galaxies refuse the cusp. High-resolution rotation curves of dwarf and low-surface-brightness galaxies, the systems where dark matter should dominate most cleanly and baryonic complications least, consistently reveal flat constant-density cores extending over substantial inner radii, a discrepancy that has reached the 5-sigma level across well-measured samples (de Blok 2010; Bullock and Boylan-Kolchin 2017). The standard repair is baryonic feedback, repeated gas blowouts gravitationally heating the cusp into a core, but the fix must work in galaxies with barely enough stars to drive it, must produce the observed diversity of core sizes without over-tuning, and must do so without breaking the other small-scale constraints. Alternative dark sectors, self-interacting, warm, fuzzy, each trade the cusp for new problems elsewhere.
The standing after three decades: the cleanest dark-matter laboratories disagree with the cleanest dark-matter prediction, and every repair complicates the simplicity that made cold dark matter compelling. WALLABY and SKA rotation curves will multiply the sample by orders of magnitude.