Transmission electron microscopy (TEM) of two-dimensional materials (2D) offers an unprecedented opportunity to study disordered systems down to the single-atom level. The reduced dimensionality of these systems provides a two-fold opportunity: first, 2D materials serve as model systems for exploring direct correlations between the structure and properties of individual atomic features. Second, these studies enable the development of new 2D materials and devices with precisely tailored optical, electronic, and mechanical properties. The experiments presented in this thesis show the first atomic-resolution images of extended one- and two-dimensional disorder in 2D materials and the extraordinary range of consequences they have on the local materials properties. The thesis begins with studies that probe the structure and properties of the 1D defects that make up grain boundaries in atomically-thin layers of graphene and molybdenum disulfide. These experiments span length scales across five orders of magnitude to image every atom at the grain boundaries through atomic-resolution scanning TEM and rapidly map the location, orientation, and shape of several hundred grains with dark-field TEM. Correlating these images with local probes of electrical, mechanical, and optical properties reveals that grain boundaries have effects that range from the unmeasurable to the extreme. A second set of projects utilizes aberration-corrected electron microscopy of a newly discovered 2D polymorph of SiO2 to conduct some of the first atomic- resolution studies of glass. Images of the atomic structure of 2D SiO2 strikingly resemble Zachariasen's foundational cartoon models of glasses and reveal distributions of medium-range ordering that will be critical for refining theoretical models for how and why glasses form. Additional experiments use the electron beam to excite and image atomic rearrangements in this 2D SiO2, producing dramatic videos that visualize the structural building blocks that control how glasses bend, break, and melt. These studies present the first data that identify the strain associated with individual ring rearrangements, observe the role of vacancies in shear deformation, and quantify fluctuations at a glass/liquid interface. In sum, this thesis demonstrates a new combination of TEM-based techniques that provide a foundation for atom-by-atom understanding and control of the macroscopic properties of emerging 2D materials and devices.