This thesis is concerned with the spectral properties of quantum particles in fractal potentials. Fractals are infinitely detailed systems that typically have non-integer dimensions. The study of these systems is motivated both by natural occurrences and possible experimental realizations of such potentials. Several spectral properties were considered for a total of thirty-three fractal potentials. Most properties displayed unique physics due to the fractal character of the systems, but not all. This thesis starts by providing a discussion on methodology and the combination of concepts from measure theory and quantum mechanics. A highly efficient method of generating fractals was developed and named the Method of Repeated Kronecker Products. In this method, the fundamental properties of the generated fractal become evident through what is referred to as the ‘generator’ of the fractal. A method for reducing the complexity of the Hamiltonian was also implemented leading to significant improvements in performance. The first property studied was the scaling of the ground state energies as the details of the fractals were gradually increased. It was found that the scaling behaved as expected for a normal Euclidean object. That is, there were no observed consequences arising from the fractal nature of the systems in these results. Next, the conductances through the fractals were evaluated. The contribution to the total conductance from the state at a given energy was considered. The conductances fluctuated rapidly when the energy was varied, and the conductances were found to be multifractal. The conductances had dimensions (capacity dimensions) equal to the fractal dimensions of the underlying potentials. These results show a clear manifestation of the fractal nature in the physically observable quantities. Furthermore, this property predicts that meaningful information can be determined for an unknown fractal structure by a conductance experiment. To better understand the observed conductance results, the localization of states was considered. This was evaluated using the participation ratios of eigenstates in the fractal potentials. The results fluctuated rapidly as a function of the eigenenergy, and it was found that the participation ratios were also fractal with dimensions equal to the fractal dimensions of the underlying potentials. This means that for a given extended state in a fractal, there exists no neighborhood of stability in energy where the state is guaranteed to remain extended. These fluctuations between extended and localized states are thought to be the cause of the observed fluctuations in the conductances. Finally, the level spacing statistics were studied. It was found that the energy level spacings of fractals with low internal connectivity followed decreasing power-law distributions for small energy spacings. This means that the likelihood of finding closely spaced eigenvalues is larger than one would expect in an open system. This might be understood as an effective energy level contraction. The result does not correspond to a periodic, aperiodic or quasiperiodic system and seems not to have a satisfactory explanation from random matrix theory