Given a cyclic group G with generator g, and given an element t in G, the discrete logarithm problem is that of computing an integer l with gl = t . The problem of computing discrete logarithms is fundamental in computational algebra, and of great importance in cryptography. In this lecture we shall examine how sometimes the problem may be reduced to the computation of discrete logarithms in smaller groups (though this reduction may not always lead to an easier problem). We give an example of how the reduction may be used profitably in taking “square roots” in cyclic groups of even order. We shall look at several exponential-time algorithms that work in a quite general setting, and we shall discuss the index calculus algorithm for taking discrete logarithms in the multiplicative group of integers modulo a prime.