Two-level quantum systems have their ubiquitous presence in chemistry and physics---in the basic ideas of bonding of atoms based on the superposition of two quantum states; in quantum optics and laser physics; in magnetic resonance; and a multitude of other phenomena. The underlying theory is well-described in many textbooks. This article aims to go beyond textbooks and familiarize university students (and hopefully excite them) about several related topics of great contemporary interest in chemistry and physics of materials, remaining however within their accessible realm. These topics are of current research activity, including tunneling centers in solids, coherent preservation of quantum information in qubits, hybridization of orbitals in carbon leading to materials such as graphene, the Berry phase, and its implication for magnetic monopoles, topological solids, and Rashba spin-split bands, etc. In discussing these themes, references are made to textbooks of quantum mechanics, but the connection is provided to advanced areas in a manner that is pedagogical and not forbidding to students.