Quantum phase transitions occur as a result of competing ground states. The focus of the present work is to understand quantum criticality and its consequences when the competition is between insulating and metallic ground states. Metal-insulator transitions are studied by means of electronic transport measurements and quantum critical points are approached by applying hydrostatic pressure in two different compounds namely Eu2Ir 22O7 and FeCrAs. The former is a ternary metal oxide and the latter is a ternary metal pnictide.Eu2Ir22O7 is a ternary metal oxide and a member of the pyrochlore iridate family. Resistivity measurements under pressure in moissanite anvil cells show the evolution of the ground state of the system from insulating to metallic. The quantum phase transition at Pc ∼ 6 GPa appears to be continuous. A remarkable correspondence is revealed between the effect of the hydrostatic pressure on Eu2Ir22O7 and the effect of chemical pressure by changing the R size in the R2Ir2O7 series. This suggests that in both cases the tuning parameter controls the t2g bandwidth of the iridium 5d electrons. Moreover, hydrostatic pressure unveils a curious cross-over from incoherent to conventional metallic behaviour at a T* > 150 K in the neighbourhood of Pc, suggesting a connection between the high and low temperature phases. The possibility of a topological semi-metallic ground state, predicted in recent theoretical studies, is explained.A major component of this work was the development of the ultra-high pressure measurements by means of Anvil cells. A novel design is introduced which minimizes the alignment accessory components hence, making the cell more robust and easier to use.FeCrAs is a ternary metal pnictide with Fermi liquid specific heat and susceptibility behaviour but non-metallic non-Fermi liquid resistivity behaviour. Characteristic properties of the compound are explained and compared to those of superconducting pnictides. Antiferromagnetic (AFM) order sets in at ∼125 K with the magnetic moments residing on the Cr site. Pressure measurements are carried out in moissanite and diamond anvil cells in order to suppress the AFM order and resolve the underlying electronic transport properties. While AFM order is destroyed by pressure, the non-metallic non-Fermi liquid behaviour is shown to be robust against pressure.