The discovery of carbon fullerene cages and their solids opened a new avenue to build materials from stable cage clusters as "artificial atoms" or "superatoms" instead of atoms. However, cage clusters of other elements are generally not stable. In 2001, ab initio calculations showed that endohedral doping of Zr and Ti atoms leads to highly stable Zr@Si-16 fullerene and Ti@ Si-16 Frank-Kasper polyhedral clusters with large HOMO-LUMO gaps. In 2002, Zr@Ge-16 was shown to form a Frank-Kasper polyhedron, suggesting the possibility of designing novel clusters by tuning endohedral and cage atoms. These results were subsequently confirmed from experiments. In the past nearly two decades, many experimental and theoretical studies have been carried out on different clusters, and many very stable cage clusters with possibly high abundance have been found by endohedral doping. Indeed in 2017, Ta@Si-16 and Ti@Si-16 cage clusters have been synthesized in bulk quantity of about 100 mg using a dry-chemistry method, giving rise to a new hope of developing cluster-based materials in macroscopic quantity besides the well-known C-60 fullerene solid. Also, wet-chemistry methods have been used to synthesize endohedrally doped clusters as well as ligated clusters and their solids, which auger well for the development of novel nanostructured materials using atomically precise clusters with unique properties. In this comprehensive review, we present results of many such developments in this fast-growing field including (i) endohedrally doped Al, Ga, and In clusters, (ii) small endohedral carbon fullerene cages with <= 28 carbon atoms, (iii) metal doped boron cages, (iv) endohedrally doped cages of group 14 elements (Si, Ge, Sn, and Pb), (v) coinage metal (Cu, Ag, Au) cages doped with a transition metal atom as well as their ligated clusters and crystals, (vi) endohedrally doped cages of compound semiconductors, and (vii) multilayer Matryoshka cages and core-shell structures. In a large number of cases, we have performed ab initio calculations to present updated results of the most stable atomic structures and fundamental electronic properties of the endohedrally doped cage clusters. We discuss electronic, magnetic, optical, and catalytic properties in order to shed light on their potential applications. The stability of the doped cage clusters has been correlated to the concept of filling the electronic shells for superatoms such as within a spherical potential model and also using various electron counting rules including Wade-Mingos rules, systems with 18 and 32 electrons, and the spherical aromaticity rule. We also discuss cluster-cluster interaction in cluster dimers and assemblies of some of the promising doped cage clusters in different dimensions. Finally, we give a perspective of this field with a bright future.