Matter at the absolute zero in temperature may reach a highly exotic state: Where two distinctly different ground states are separated by a second order phase transition the system is far from being frozen; it is undecided in which state to be and therefore undergoes strong collective quantum fluctuations. Quantum criticality describes these fluctuations and their extension to finite temperature. Quantum critical behaviour has been reported in systems as distinct as high-temperature superconductors, metamagnets, multilayer $^3$He films, or heavy fermion compounds. The latter have emerged as prototypical systems in the past few years. A major puzzle represents the recent discovery of a new energy scale in one such system, that vanishes at the quantum critical point and is in addition to the second-order phase transition scale. Completely new theoretical approaches are called for to describe this situation. In this project we want to explore the nature of this new low-lying energy scale by approaches that go significantly beyond the state-of-the-art: apply multiple extreme conditions in temperature, magnetic field, and pressure, use ultra low temperatures in a nuclear demagnetization cryostat, and perform ultra-low energy spectroscopy, to study carefully selected known and newly discovered heavy fermion compounds. Samples of outstanding quality will be prepared and characterized within the project and, in some cases, be obtained from extrenal collaborators. New approaches in the theoretical description of quantum criticality will accompany the experimental investigations. The results are likely to drastically advance not only the fields of heavy fermion systems and quantum criticality but also the current understanding of phase transitions in general which is of great importance far beyond the borders of condensed matter physics.