Quantum information theory is the study of communication and information processing tasks using physical systems that obey the rules of quantum theory. Information theory was largely the creation of Claude Shannon working at Bell laboratories over fifty years ago. Shannon produced an elegant mathematical theory for information encoding, transmission and decoding in the presence of noise. In the early 1980’s a number of pioneers including, Holevo, Schumachor, Brassard, Ekert and Bennett, began to re-consider these issues in the light of quantum noise. In this chapter we will consider some of these developments including quantum cryptography and quantum teleportation.