Two-dimensional layered crystals are a promising class of materials for post-silicon electronics. Due to their atomic thinness, flexibility, and versatile electrical properties (i.e. conductors, semiconductors, and insulators), we can envision future ultra-small, flexible computers completely comprised of various two-dimensional materials. For this application, lateral heterostructures of two-dimensional materials play a major role in the realization of wholly two- dimensional devices, as they are the fundamental elements in a circuit, such as p-n junctions and metal-semiconductor contacts. This dissertation will employ transmission electron microscopy and related techniques to address how different two-dimensional materials merge to form lateral heterostructures, specifically between two distinct two-dimensional semiconductors (analogous to p-n junctions) and two-dimensional conductor- semiconductor heterostructures (analogous to metal-semiconductor contacts). Within the heterostructures between two semiconductors, Chapter 2 and 3 will discuss atomically sharp interfaces and gradual interfaces in lateral heterostructures, respectively. Chapter 4 will describe the conductor-semiconductor interconnects between two-dimensional materials with dissimilar lattice structures. Our results demonstrate how the strain is relaxed in epitaxial lateral heterostructures, as well as how the heterostructure between crystallographically distinct two-dimensional materials forms. These findings can unravel how to use or engineer distortions in two-dimensional lateral heterojunctions, predict the mechanical strength and devices performance, and inform the mechanism of chemical synthesis at the interface between atomically thin films.