Quantum criticality is a new subdiscipline of condensed-matter physics. It deals with so-called quantum critical points (QCPs) at which different ground states (e.g., magnetic and nonmagnetic, electrically conducting and insulating) compete with each other. QCPs give rise to exotic finite-temperature properties. They often also promote the formation of novel phases, notably unconventional superconductivity . Heavy-fermion (HF) compounds or Kondo-lattice systems have emerged as prototypical materials to study quantum criticality. In HF compounds where the QCP is tuned, e.g., by pressure or magnetic field, the two competing ground states are normally an antiferromagnetically ordered and a nonmagnetic, metallic (a so-called Landau-Fermi-liquid) state. On the other hand, the existence of ferromagnetic QCPs in Kondo-lattice systems is as yet not settled. In the MPI-CPfS, a novel type of QCP is currently studied where an antiferromagnetic instability coincides with an orbital-selective Mott transition (or Kondo breakdown QCP). The latter may be detached from the former, e.g., by suitable chemical pressure. In this way, a reconstruction of the Fermi surface and, eventually, a transition from the well-established Landau Fermi-liquid phase to a novel spin-liquid phase may be observable in the absence of magnetic order.