Bose-Einstein condensates (BEC) of exciton-polaritons display macroscopic quantum phenomena at elevated temperatures on a robust solid-state platform. However, in contrast to its atomic counterparts, an exciton-polariton BEC exhibits non-equilibrium behaviour arising from its open-dissipative nature and coupling to a thermal reservoir. Thus, despite the advancement in both theory and experiments, there are still major issues in our understanding of this system. In this PhD Final seminar, I will present a series of experiments aimed to better understand the system by looking at its behaviour in different trapping potentials. In particular, we demonstrate experimental control over the occupation and the complex eigenvalues of the potentials by structuring the excitation profile. This leads to our ability to load condensates into distinct Bloch states of periodic potentials, and visualise Berry phase and observe exceptional points in non-Hermitian exciton-polariton resonators. Furthermore, using single-shot condensate imaging, we observe clear signatures of strong reservoir depletion, which are hidden in typical ensemble-averaged experiments. Finally, by utilising the effect of the reservoir depletion, we create a round "box" trap for exciton-polaritons. The condensate confined in this trap is characterized by a Thomas-Fermi density distribution, single-mode ground state occupation, and large energy blueshift, allowing us to perform direct and accurate measurement of the polariton-polariton interaction strength.