When waves propagate in a medium in which the microstructure is much smaller than the wavelength, the waves behave almost as if they were propagating in a homogeneous medium characterized by effective moduli. In this lecture I will discuss and prove the analytic properties of the effective moduli as a function of the component moduli in a two or multicomponent composite. The analyticity properties have important ramifications. They provide a justification for the correspondence principle, and enable one to place bounds on the effective moduli. These bounds hold for all conceivable microstructures.