The objective of this chapter is to introduce the broad range of concepts necessary to appreciate and understand the various aspects and properties of phononic crystals and acoustic metamaterials described in subsequent chapters. These concepts range from the most elementary concepts of vibrational waves, propagating waves, and evanescent waves, wave vector, phase and group velocity, Bloch waves, Brillouin zone, band structure and band gaps, and bands with negative group velocities in periodic or locally resonant structures. Simple models based on the one-dimensional harmonic crystal serve as vehicles for illustrating these concepts. We also illustrate the application of some of the tools used to study and analyze these simple models. These analytical tools include eigenvalue problems (ω(k) or k(ω)) and Green’s function methods. The purpose of this chapter is primarily pedagogical. However, the simple models discussed herein will also serve as common threads in each of the other chapters of this book.