A grand challenge is afoot to characterize the novel two-dimensional (2D) materials that are being produced in increasing types and complexities. In particular, in 2D materials, characterizing each and every atom is critical; Rogue atoms, like defects and dopants, can have dramatic effects on the properties of materials that are a single unit-cell thick. Transmission electron microscopy (TEM) has emerged as a powerful and flexible tool to characterize the structure and properties of 2D membranes from the micron to atomic scales. Using TEM techniques to study the growth of 2D crystals, such as graphene and molybdenum disulfide, and even image the atoms at the grain boundaries. Further, by correlating TEM to ex-situ measurements, we can probe the mechanical, electrical, and optical properties of defects and grain boundaries. These studies are critical to optimizing the synthesis of a growing array of 2D materials with wide-ranging applications, including flexible electronics and layered photovoltaic devices. In addition to their practical applications, 2D materials are ideal platforms for novel TEM studies at the single-atom scale. By studying a new 2D silica glass, we show that it is now possible to image atoms in disordered solids and track their motions in response to local strain. These studies provide new insight to the mechanisms of plastic deformation, which unlike in crystals, are not well understood in amorphous materials.