NONTECHNICAL SUMMARY Two-dimensional materials are composed of atomically thin sheets of atoms that can be manipulated with an unprecedented level of control. For example, in addition to being able to isolate and stack these atomically thin materials, it is now possible to twist a pair of them to within 0.1 degrees of accuracy. As a result, new arrangements of atoms form when placed on top of each other in this fashion that would not form in nature. At the same time, varying the density of electrons in these devices changes the system from a semiconductor to a superconductor, metal, or insulator providing an amazing amount of control and emergence not previously seen in a single device. These advances raise several new fundamental questions that need to be theoretically and computationally addressed to explain the experimental observations. The amazing tunability of these systems makes them promising candidates for the construction and application of novel technological devices that can be used in future electronics and atomic-scale measurement devices. This research project takes into account realistic effects of the twisted materials to understand how they impact the overall response of the electronic system to imperfections and randomness in the materials, to develop new probes to be used in experiment to ascertain physical effects that are currently out of reach, as well as to find new phases of matter that have yet to be discovered that result from bringing these