The work presented in this dissertation has been focused on structure, stability, electronic properties, and chemical bonding of atomic clusters and solid-state compounds. The common thread was development of chemically intuitive models and theoretical methods capable of describing and interpreting bonding and hence, structures of these compounds. Understanding how interactions between atoms in sub-nano clusters and solid-state compounds of certain compositions determine their structures, physical properties, and reactivities is essential for rational design of new materials, catalysts, and molecular devices. A significant part of this work presents joint experimental and theoretical studies of doped boron clusters. Several projects on carbon- and aluminum-substituted boron clusters were aimed at establishing their structures, energetic and electronic properties, and understanding bonding interactions. The dissertation introduces a series of peculiar clusters containing transition metal atoms inside perfectly symmetrical boron rings. These clusters, featuring planar octa-, nona-, and decacoordinated transition metal atoms, were designed based on a simple chemical bonding model governing stabilities of such species. One of the most important parts of this dissertation deals with chemical bonding in the solid state. The Adaptive Natural Density Partitioning method previously developed by the Boldyrev group at Utah State University has proven very efficient for understanding chemical bonding in clusters and complex molecules. In this work, a periodic implementation of this method has been developed, yielding a new theoretical tool capable of interpretation of bonding in solid state in chemically intuitive terms of localized and multi-center bonds.