While not telling the whole story, the representation of a composite medium by a homogeneous effective medium is often an excellent approximation for describing its macroscopic physical properties. Modern methods for calculating the effective medium properties are reviewed with special emphasis on understanding both successes and limitations. Outstanding problems that can and should be tackled are identified. The successes include calculations of the electrical conductivity, dielectric coefficient, and elastic stiffness moduli of composites with a periodic microstructure, and the simulation of those properties for disordered composites near a percolation threshold by means of discrete models such as a random-resistor-network. For composites where the microstructure is either unknown or very complicated, a whole class of exact bounds have been found for these properties based on various types of limited information. Recently, advances have been made in calculating the weak field magneto-transport and the thermoelectric behavior of two-component composites, and also some types of nonlinear properties. An important challenge remains the calculation of magneto-transport at high magnetic fields. Another is the theoretical treatment of multicomponent composites. A third is to find relations between different effective properties of a composite that can enable us to learn about property A by measuring a different property B. This is especially important when the measurement of A would destroy the sample, as when A is the yield stress, whereas the measurement of B is nondestructive, as when B is a small, nonlinear correction to the usual elastic stiffness moduli.