Quantum phase transitions are among the most intriguing topics in modern condensed matter physics. Interesting physical phenomena usually emerge in the vicinity of quantum critical points, making the investigation of such quantum systems particularly rewarding. In this thesis we present studies related to quantum phase transitions in two compounds, FBBO and Sr3Ru2O7, using hydrostatic pressure as the tuning parameter. Pressure is an ideal parameter to tune the strength of various interactions within the sample by changing the lattice constants, but is known to be very challenging to apply experimentally. Therefore a major effort of my PhD is to find out a consistent and systematic approach to prepare such experiments. FBBO is a neutral organic radical that has been carefully designed to be as metallic as possible at ambient pressure. Nevertheless it is still insulating, and high pressure was required in order to drive this system through the metal-insulator transition. Here, I present strong evidence of the formation of a Fermi liquid ground state under 6.2 GPa [1]. This is the first such observation for a neutral organic radical, after being proposed for decades. In the approach to metallization, we also found intriguing evidence of a low temperature magnetic phase. The c-axis resistivity of the itinerant metamagnet Sr3Ru2O7 has also been measured under pressure. The initial goal was to search for superconductivity, largely because of the interesting unconventional superconducting ground state found in its close sibling, Sr214. In particular, the c-axis conductivity was measured to rule out possible Sr2RuO4 inclusions within the sample. No superconductivity was found up to the highest pressures of 5.8 GPa and 4.7 GPa respectively in the two independent sets of measurements. Instead, we found that the high temperature resistance falls substantially with increasing pressure, unlike what has been reported for Sr214. The metamagnetic transition shifts rapidly with pressure to higher magnetic field. More importantly, power-law analysis at low temperatures at various pressures indicates that the strength of the electron-electron interaction decays rapidly with increasing pressure. This offers a natural explanation for our failure to induce superconductivity at high pressures.