In this chapter, the underlying science of fibres and matrices is described. Some specific examples are given to illustrate the key factors involved. A wide range of reinforcements, mostly in the form of fibres, is now available commercially. Their properties can be related directly to the atomic arrangement and the defect content of the reinforcement, which must be controlled in the manufacturing processes. Matrices may be based on polymers, metals or ceramics. The choice of matrix is related to the required properties, the intended applications of the composite and the method of manufacture. The properties of the matrix depend on microstructure which, in turn, depends on manufacturing route and subsequent thermal and mechanical treatments. Certain properties of the composite may be sensitive to the nature of the reinforcementi/matrix interface; this topic is covered in detail in Chapter 7.