Self-assembled monolayers are a unique class of nanostructured materials, with properties determined by their molecular lattice structures, as well as the interfaces with their substrates and environments. As with other nanostructured materials, defects and dimensionality play important roles in the physical, chemical, and biological properties of the monolayers. In this review, we discuss monolayer structures ranging from surfaces (two-dimensional) down to single molecules (zero-dimensional), with a focus on applications of each type of structure, and on techniques that enable characterization of monolayer physical properties down to the single-molecule scale.