Recent advances in solid state materials engineering have led to mesoscopic devices with feature sizes that approach the fundamental quantum wavelength of charge carriers in the solid, allowing for the experimental observation of quantum interference. By confining carriers to a single quantum state in one or more dimensions, the degrees of freedom for charge transport can be reduced to achieve new device functionality. This dissertation focuses on mesoscopic electron billiards that combine the aspects of zero, one, and two-dimensional transport into one system. Low-temperature measurement of billiards fabricated within a relatively defect-free semiconductor heterostructure results in ballistic transport, where the electron waves follow classical trajectories and the confining walls play a major role in determining the electron interference. Billiards have been traditionally formed by applying a bias to patterned surface gates atop an AlGaAs/GaAs heterostructure. Within this system, fractal fluctuations in the billiard conductance are observed as a function of an applied external magnetic field. These fluctuations are tied to quantum interference via an empirical parameter that describes the resolution of energy levels within the billiard. To investigate whether fractal fluctuations are a robust phenomenon intrinsic to billiard-like structures, this study centers on billiards defined by etching walls into a GaInAs/InP heterostructure, departing from the traditional system in both the type of confinement and material system used. It is expected that etched walls will provide a steeper confinement profile leading to well-defined device shapes. Conductance measurements through the one-dimensional leads that couple electrons into the billiard are utilized in combination with a self-consistent Schrodinger/Poisson solution to demonstrate a steeper confinement potential. Experiments are also carried out to determine whether fractal fluctuations persist when billiards are coupled together to form arrays. While fractal scaling is observed in solitary etched billiards, conditions arise in which the fluctuations depart from fractal scaling in both single billiards and arrays. An analysis of the phase-breaking time governing quantum interference reveals a fundamental transition in the interference behavior between single billiards and arrays. Concluding remarks discuss the possibility of the observation of fractal fluctuations in nanoscale particles at higher temperatures.