Diffie and Hellman’s famous 1976 paper, ”New Directions in Cryptography,” lived up to its title in providing the directions that cryptography has followed in the past thirty years. Where will, or should, cryptography go next? This talk will examine this question and consider many possible answers including: more of the same, number-theoretic algorithms, computational-complexity approaches, quantum cryptography, circuit-complexity methods, and new computational models. Opinions will be offered on what is most likely to happen and what could be most fruitful. These opinions rest not on any special competence by the speaker but rather on his experience as a dabbler in, and spectator of, cryptography for more than forty years.