Chapter 1 reviews the theoretical principles necessary for understanding the remaining chapters of the book and also reviews the fundamentals of quantum mechanics and electronic band structures. Physical properties of materials on nanometer scale are governed by quantum mechanics. The rst part of this chapter describes the fundamentals of quantum physics, including the uncertainty principle, the Schrödinger equation, matrix mechanics, and perturbation theory. The electronic structure of materials is determined by crystallographic periodicity, which is represented by the electronic band structure. In the second part of this chapter, the concept of electronic band structure is examined based on the nearly free electron approximation and the tight binding model. As an introduction to other chapters in the book, the nal part of this chapter describes the effect of size on the electronic, optical, and magnetic properties of materials, with an emphasis on the nanometer scale and properties of nanomaterials.