When electrons are confined to a two dimensional layer with a perpendicular applied magnetic field, such that the ratio of electrons to flux quanta \((\nu)\) is a small integer or simple rational value, these electrons condense into remarkable new phases of matter that are strikingly different from the metallic electron gas that exists in the absence of a magnetic field. These phases, called integer or fractional quantum Hall (IQH or FQH) states, appear to be conventional insulators in their bulk, but behave as a dissipationless metal along their edge. Furthermore, electrical measurements of such a system are largely insensitive to the detailed geometry of how the system is contacted or even how large the system is... only the order in which contacts are made appears to matter. This insensitivity to local geometry has since appeared in a number of other two and three dimensional systems, earning them the classification of "topological insulators" and prompting an enormous experimental and theoretical effort to understand their properties and perhaps manipulate these properties to create robust quantum information processors. The focus of this thesis will be two experiments designed to elucidate remarkable properties of the metallic edge and insulating bulk of certain FQH systems. To study such systems, we can use mesoscopic devices known as single electron transistors (SETs). These devices operate by watching single electrons hop into and out of a confining box and into a nearby wire (for measurement). If it is initially unfavorable for an electron to leave the box, it can be made favorable by bringing another charge nearby, modifying the energy of the confined electron and pushing it out of the box and into the nearby wire. In this way, the SET can measure nearby charges. Alternatively, we can heat up the nearby wire to make it easier for electrons to enter and leave the box. In this way, the SET is a sensitive thermometer. First, by operating the SET as an electrometer, we measure the local charge of the \(\nu = 5/2\) FQH state. An immediate consequence of measuring fractionally quantized conductance plateaus is that the charge of local excitations should be a fraction of \(e\), the charge of an electron. The simplest charge that would be expected at \(\nu = 5/2\) would \(e/2\). However, if the charged particles that condense into the \(\nu = 5/2\) FQH state are paired, the expected local charge becomes \(e/4\). By watching these local charges being added to compressible puddles at \(\nu = 5/2\) and \(\nu = 5/7\), we find that the local charge at \(\nu = 5/2\) is indeed \(e/4\), indicating that objects of charge \(e\) are pairing to form the ground state of the system. This has implications for the future possibility of detecting non-Abelian braiding statistics in this state, and is described in detail in Chapter 2. By further monitoring how eagerly these \(e/4\) particles enter puddles as we increase the temperature, we can attempt to identify the presence of some excess entropy related to an unconventional degeneracy of their ground state. Such an entropy would be expected if the \(\nu = 5/2\) state exhibited non-Abelian braiding statistics. Progress on these experiments and prospects for building a quantum computer are presented in Chapter 3. Next, by operating the SET as a thermometer, we monitor heat flow along the compressible edge and through the bulk of IQH and FQH states. As an edge is heated and charge on that edge is swept downstream by the external magnetic field, we expect that charge to carry the injected energy in the same downstream direction. However, for certain FQH states, this is not the case. By heating an edge with a quantum point contact (QPC) and monitoring the heat transported upstream and downstream, we find that heat can be transported upstream when the edge contains structure related to \(\nu = 2/3\) FQH physics. Surprisingly, this can be present even when the bulk is in a conventional insulating (IQH) state. Additionally, we unexpectedly find that the \(\nu = 1\) bulk is capable of transporting heat, while the \(\nu = 2\) and \(\nu = 3\) bulk are not. These experiments are presented in Chapter 4. Finally, in Chapter 5, we describe preliminary work on a very different type of topological material, the quantum spin Hall (QSH) insulator. Here, the spin of electrons takes the place of the external magnetic field, creating edge states that propagate in both directions. Each of these edges behaves as an ideal one-dimensional mode, with predicted resistance \(h/e^2\). By creating well-defined regions where these modes can exist, we identify and characterize the conductance associated with topological edges. / Physics