This chapter describes that isolated atomic and molecular energy levels broaden into bands of allowed energy states in solids, separated by forbidden energy gaps. It considers in more detail the structure of these allowed energy bands. The chapter introduces Bloch's theorem, and describes how its application considerably simplifies the calculation and description of the electronic structure of crystalline solids. It is highly surprising that the nearly free electron model should ever work in solids, as the electron-nucleus interaction has very sharp singularities, where the potential deviates very strongly from the flat, free space potential. The chapter illustrates many of the basic properties of electrons in a periodic solid by using the Kronig-Penney (KP) model. It outlines how the tight-binding (TB) method can be used to successfully calculate the band structure of the K-P model from 'first principles'. The chapter shows that the TB method works best for the lowest lying energy levels, becoming less accurate for higher-lying excited states.