Self-assembled nanoscopic metamaterials are a class of powerful new materials for a wide variety of applications, due in part to their highly tunable functionalities and structures. Binary colloidal crystals, in which two different types of colloidal particles are incorporated into one ordered structure, show greater potential for structural diversity than single-component colloidal crystals. Consequently, the diverse electronic, magnetic, and optical properties and potential applications of colloidal crystals make them promising candidates for advancing numerous technologies and addressing key challenges in fields ranging from novel devices and photonics to healthcare and environmental sustainability. This thesis investigates the self-assembly of bidisperse systems of nanoparticles into colloidal crystals in both two and three dimensions, in systems mimicking both metallic alloys and salts. In the first project, the influence of increasingly disparate sizes of particles on the stability of mixed, bcc-type crystals is explored, finding that the addition of a second characteristic interaction length scale stabilizes mixed bcc crystals in systems with larger size dispersities. In the second project, the robustness of a variety of different crystal structures with respect to size dispersity is investigated in bidisperse systems with a tunable particle--particle interaction model. In the third project, a phase diagram is constructed for binary systems of particles interacting with "ionic"-like interactions in two dimensions. All of these studies are conducted with molecular dynamics simulations, and employing abstract model systems of point particles interacting via isotropic pair potentials. The simplicity of the computational models used---one- and two-well Lennard-Jones--Gauss pair potentials in the first and second study, and only Lennard-Jones and Weeks--Chandler--Andersen pair potentials in the third study---allow for a high degree of transferability of these findings to various specific particle systems and materials. The insights gained into the assembly of bidisperse and binary structures in the presence of a variety of dispersities (and stoichiometries) will enable a targeted selection of assembled structures and tailored interactions with the aim of achieving robust self-assembly processes in realistic particle systems.