For simplicity, calculations in seismic wave theory often assume that the Earth is perfectly elastic. But the real Earth is anelastic, meaning that wave energy is absorbed by internal friction effects. This chapter gives an introductory account of the mathematical theory of seismicwave propagation in anelastic media. The important concepts of the quality factor (Q), the loss factor (1/Q) ,and the complex modulus are introduced. The necessity for including dispersion in computations of the shapes of waveforms in anelastic media is demonstrated. Spring-dashpot models for describing anelastic media, and the more general linear theory of viscoelasticity, are introduced. The nature of Q in the Earth is discussed. The 1D and 3D equations of motion in anelastic media, and their plane wave solutions, are derived. The concept of general plane waves (including homogeneous and inhomogeneous plane waves) is introduced.Reflection and transmission of plane waves, as well as particle motion, in anelastic media are briefly discussed. The theory of exactly constant Q is briefly covered.