Our aim in this article is to apply Darwin’s idea of natural selection to the maturation of the immune response. In so doing, we shall introduce recent results concerning the character of adaptive walks via fitter mutants to local optima, where such walks occur in rugged fitness landscapes with many peaks, ridges, and valleys. In the present context, the fitness landscape in question is determined by the affinity with which each possible antibody molecule in an organism’s repertoire binds to a given immunizing epitope. By an adaptive walk, we mean a random walk on the affinity landscape where a step can only be taken if it leads to a higher affinity. Maturation of the immune response refers to the observation that after immunization the average affinity of serum antibody for the immunizing epitope increases.