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Quasicrystals are a class of materials that are long-range ordered yet not periodic. These unique features make them a fascinating middle ground between order and disorder, thus providing an ideal platform for studying a wealth of physical phenomena. This thesis contains three experimental works that my colleagues and I achieved during my PhD period, in which we explored the properties of quasicrystals in many aspects. The first experiment (Chapter 5) focus on the long-range order in quasicrystals. We report on the first experimental realisation of a two-dimensional (2D) quasicrystalline optical lattice for ultracold atoms and conduct matter-wave diffraction experiment using shot, intense lattice pulses. We reveal the different diffraction dynamics between periodic and quasiperiodic lattices and demonstrate the capability to simulate quantum walks on four-dimensional tight-binding lattices using the diffraction dynamics of 2D quasicrystalline lattices on short timescales. The second (Chapter 6) and the third (Chapter 7) experiments concentrate on studying the disorder nature of quasicrystals. In particular, we investigate disorder-induced localisation transition in the ground state of non-interacting and weakly interacting Bose gases in a 2D quasicrystalline lattice. The second experiment is mainly performed in the non-interacting limit, in which we probe the localisation transition by employing triangular lattice pulses and studying the time scale required for adiabatic loading. We observe a localisation transition at around a critical lattice depth of Vloc ≈ 1.78 Erec for noninteracting systems. In addition, we demonstrate that weak repulsive interactions can shift transition to deeper lattices. In the third experiment, we further study the interplay between the interactions and disorder, and constitute the first experimental realisation of the 2D Bose glass, an insulating but compressible groundstate phase without long-range phase coherence. By probing the coherent properties of the system, we observe a Bose glass to superfluid transition and map out the phase diagram in the weakly interacting regime. Moreover, we reveal the non-ergodic nature of the Bose glass by probing the capability of restoring coherence. Our observations are in good agreement with recent quantum Monte Carlo predictions and pave the way for experimentally testing of the connection between the Bose glass and many-body localisation.