The work presented in this thesis investigates the behaviour of the Rashba semi-conductor BiTeI and of the topological insulator $Bi2Te3$ under pressure. Using Shubnikov-de Haas quantum oscillation measurements, the evolution of the Fermi surface of both materials was tracked as a function of pressure.

At ambient pressure, two distinct quantum oscillation frequencies in BiTeI, corresponding to inner and outer Fermi surface orbits as a result of spin-splitting caused by the Rashba effect, were observed. Using a model Hamiltonian with a Rashba interaction term to model this system, experimental results were fitted to determine model parameters. Based on this model, carrier densities for the samples were calculated and there was good agreement with Hall effect measurements. The phase of the oscillations showed that both Fermi surfaces have a Berry phase of $\pi$ associated with them, consistent with theoretical predictions for a Rashba system.

As pressure is applied, it was observed that the inner Fermi surface expands while the outer Fermi surface shrinks. Phase analysis of the oscillations showed deviations from the ambient pressure value, hinting at a topological transition.

For $Bi2Te3$, we report the observation of two oscillation frequencies ($\sim 40$ T and $\sim 340$ T) at ambient pressures. Based on the angular dependence of the oscillation frequencies, phase analysis, and comparison against band structure from published ARPES results, it is deduced that the higher frequency oscillation corresponds to the surface state of $Bi2Te3$. Non-linear behaviour in the Hall measurement also suggests the presence of multiple bands, and a two-band model with parameters derived from quantum oscillation measurements is used to fit the experimental data.

Under pressure, a slight decrease in the low field Hall coefficient and a new frequency appearing at $\sim 20$ kbar was observed. These may be signatures of a change in the Fermi surface of $Bi2Te3$ caused by an electronic topological transition.